We obtain a localized version of the configuration space integral for the Casson knot invariant, where the standard symmetric Gauss form is replaced with a locally supported form. An interesting technical difference between the arguments presented here and the classical arguments is that the vanishing of integrals over hidden and anomalous faces does not require the well known “involution tricks”. Further, the integral formula easily yields the well-known arrow diagram expression for the
invariant, first presented in the work of Polyak and Viro. We also take the next step of extending the  arrow diagram expression to multicrossing knot diagrams and obtain a lower bound for the übercrossing number. The primary motivation is to better understand  a connection between the classical configuration space integrals and the arrow diagram expressions for finite type invariants.

§ INTRODUCTION

Configuration space integrals,  originally introduced by Bott and Taubes in [8] and motivated by the work of Kontsevich [22, 23], are a far-reaching generalization of the celebrated Gauss linking number
integral formula [18] for a $2$–component link.  Given a smooth embedding of $S^1\sqcup S^1$ into $\R^3$, $L=(L_1, L_2):S^1\sqcup S^1\longrightarrow \R^3$, the linking number can be computed as:

\begin{equation}\label{eq:lk-number-integral}
\operatorname{lk}(L_1,L_2)=\varint\limits_{S^1\times S^1} h^\ast_{2,1} \omega=\frac{1}{2\pi} \varint^{1}_0 \varint^{1}_0 \frac{\langle \dot{L}_1(s), \dot{L}_2(t), L_2(s)-L_1(t)\rangle}{|L_2(s)-L_1(t)|^3} ds\,dt,
\end{equation}

where $h_{12} :S^1\times S^1\longrightarrow S^2$ denotes the Gauss map of $L$: $h_{12}(s,t)=\frac{L_2(s)-L_1(t)}{|L_2(s)-L_1(t)|}$, and $h^\ast_{12} \omega$ the pullback of the
rotationally symmetric  unit area from  $\omega$ over $S^2$.
Note that the domain $S^1\times S^1$ can be regarded as the ordered configuration space $\Cnf(S^1\sqcup S^1; 1,1)$
of a pair of points in $S^1\sqcup S^1$, with the first and second point restricted to the first and second factor in $S^1\sqcup S^1$ respectively.  Recall, given a topological space $X$, the configuration space of $n$ points in $X$ is the “deleted” product

\[
\Cnf(X;n)=\{(x_1,\ldots,x_n)\in X^n\ |\ x_i\neq x_j \  \text{for} \ i\neq j\}.
\]

§.§ Integral for the Casson knot invariant
General configuration space integrals for knots and links were investigated in number of works: [8, 11, 24, 32, 33], and involve integrals over various pieces of configuration spaces of points on the knot and off the knot (these pieces are labeled by trivalent diagrams).
Besides the linking number, one of the simplest such integrals computes the second coefficient of the Conway polynomial $c_2=c_2(K)$, [16] of a closed knot $K:I\longrightarrow \R^3$, $I=[0,1]$, $K(0)=K(1)$, which is also known as the Casson knot invariant, [29]. In the following, we denote the space of classical pointed knots by $\mathscr{K}_{S^1}$, i.e. the smooth pointed embeddings of $S^1$ in $\R^3$. It was first rigorously established in the works of Bott and Taubes [8], Bar-Natan [7], and later generalized in [32, 33], the integral reads as follows

\begin{equation}\label{eq:casson-integral-original}
\begin{split}
c_2(K)  & = \frac{1}{4}\varint\limits_{\Cnf(I;n)} \omega_{13}\wedge\omega_{24}-\frac{1}{3}\varint\limits_{\Cnf(I,K;3,1)} \omega_{1\mathbf{4}}\wedge\omega_{2\mathbf{4}}\wedge\omega_{3\mathbf{4}},
\end{split}
\end{equation}

where[Note that these are connected components of the respective open configuration spaces, and we use the same notation $\Cnf$ for them.]

\begin{equation}\label{eq:connected-comp-conf}
\begin{split}
\Cnf(I;4) & \cong\bigl\{(x_1,x_2,x_3,x_4)\in I^4\ |\ 0\leq x_1<x_2<x_3<x_4\leq 1\bigr\},\\
\Cnf(I,K;3,1) & \cong\bigl\{(x_1,x_2,x_3,K,\mathbf{x})\ |\ (x_1,x_2,x_3)\in I^3; 0\leq x_1<x_2<x_3\leq 1; \\
& \qquad\qquad\qquad\qquad\quad  \mathbf{x}\in \R^3-\{K(x_1), K(x_2), K(x_3)\}; K\in \mathscr{K}_{S^1} \bigr\},
\end{split}
\end{equation}

and integral terms in (<ref>) can be explicitly written as follows

\begin{multline*}
\varint\limits_{\Cnf(I;4)} \omega_{13}\wedge\omega_{24}=\frac{1}{(4\pi)^2} \varint\limits_{\Cnf(I;4)} \prod^2_{i=1}\frac{\bigl\langle K(x_{i+2})-K(x_i),\dot{K}(x_i),\dot{K}(x_{i+2})\bigr\rangle}{|K(x_{i+2})-K(x_i)|^3}  dx_1 dx_2 dx_3 dx_4,\\ \varint\limits_{\Cnf(I,\K_{S^1};3,1)} \omega_{1\mathbf{4}}\wedge\omega_{2\mathbf{4}}\wedge\omega_{3\mathbf{4}}=\frac{1}{(4\pi)^3}\varint\limits_{\Cnf(I,\K_{S^1};3,1)} \operatorname{det}\bigl(E_1(x_1,\mathbf{x}),E_2(x_2,\mathbf{x}),E_3(x_3,\mathbf{x})\bigr) dx_1 dx_2 dx_3 d\mathbf{x},
\end{multline*}

where $E_i(x_i,\mathbf{x})=\tfrac{(K(x_i)-\mathbf{x})\times \dot{K}(x_i)}{|K(x_i)-\mathbf{x}|^3}$, $i=1,2,3$. As in the case of the linking number, the $2$–forms $\omega_{ij}$ in (<ref>) are obtained as pullbacks of the standard unit area form $\omega$ on $S^2$, along the Gauss maps:

\[
\tilde{h}_{ij}(x_i,x_j)=\frac{K(x_j)-K(x_i)}{|K(x_j)-K(x_i)|},\qquad \tilde{h}_{i\mathbf{4}}(x_i,\mathbf{x})=\frac{\mathbf{x}-K(x_i)}{|\mathbf{x}-K(x_i)|}.
\]

One immediate issue in establishing integrals of type (<ref>) is addressing the divergence of forms $\omega_{ji}$ as $x_i$ and $x_j$ or $\mathbf{x}$ approach each other.
This is achieved by replacing the domains of integration $\Cnf(I;4)$ and $\Cnf(I,K;3,1)$ with their compactifications $\Cnf[I;4]$ and $\Cnf[I,K;3,1]$ (Appendix <ref>). In order to simplify our considerations, in place of closed knots $\K_{S^1}$, we consider the space of long knots $\K=\K_{\R}$ [29, 33, 24], i.e. smooth embeddings $K:\R\longrightarrow \R^3$ of $\R$ into $\R^3$, which are standard outside some box, e.g. the image of $K$: $\{K(s)\ |\ s\in \R\}$ is contained in the cube $-1\leq x,y,z\leq 1$ in $\R^3$ for $-1\leq s\leq 1$ and equal to the $x$–axis outside the cube $s\geq 1$ and $s\leq -1$, see Figure <ref>. The integral (<ref>) can be defined analogously for long knots, by replacing the configuration spaces in (<ref>) over $S^1$, with the ones over $\R$ and appropriately adjusting the coefficients in (<ref>), which is further reviewed in Section <ref>.

Plane diagrams for the figure 8 long knot (right) and figure 8 knot (left)

§.§ From integrals to combinatorial formulas
An interesting property of the Gauss linking number integral (<ref>) is that it easily converts to
a combinatorial formula for $\operatorname{lk}(L_1,L_2)$. Indeed, (<ref>) is an integral expression for the degree of the Gauss map $h_{12}$.
Therefore, whenever $L$ has a regular planar projection[i.e. with transverse double crossings.] (onto the plane orthogonal to $N=(1,0,0)$), the local degree formula [20] applied to the Gauss map parametrized by $h_{12}:(s,u)\xrightarrow{L} (L_1(s),L_2(u))$ and the regular value $N$, yields a signed crossing count formula for $\operatorname{lk}(L_1,L_2)$:

\begin{equation}\label{eq:lk-crossing-count}
\operatorname{lk}(L_1,L_2)=\deg(h_{12})=\sum_{s\in h^{-1}_{12}(N)} \operatorname{sign}(D h_{12}(s))=\sum_{s\in \operatorname{Cross}(L)} \operatorname{sign}(s),
\end{equation}

$\operatorname{Cross}(L)$ is the set of overcrossings, $L_2$ over $L_1$.

A positive crossing (left) and negative crossing (right).

It is a classical argument, c.f. <cit.>, that the combinatorial formula (<ref>) can be obtained directly from the integral (<ref>),  by replacing the area form $\omega$ on $S^2$ (the generator of $H^2(S^2)$) with a unit area bump form $\eta_N$, supported on an $\varepsilon$–disk $D^2_\varepsilon\subset S^2$, centered at the regular value $N$ of $h_{12}$, i.e.

\begin{equation}\label{eq:eta_N}
\eta_N=\eta^\varepsilon_N=f_{N,\varepsilon}\omega,\qquad \supp(f_{N,\varepsilon})\subset D^2_\varepsilon,\qquad \varint_{S^2} \eta_N=1,
\end{equation}

where $f_{N,\varepsilon}$ is just a real valued bump function supported on the disk $D^2_\varepsilon\subset S^2$ around $N$.
The $2$–form $\eta_N$ is not anti-symmetric i.e.

\begin{equation}\label{eq:antipodal-eta_N}
A^\ast \eta_N=-\eta_S,
\end{equation}

under the antipodal map $A:\xi\longrightarrow -\xi$ of $S^2$,
where $S$ is the antipodal point of $N$.
Indeed, substituting $\omega=d\alpha+\eta_N$ Stoke's theorem yields

\begin{equation}\label{eq:lk-eta_N}
\operatorname{lk}(L_1,L_2) =\varint\limits_{S^1\times S^1} h^\ast_{12} \omega=\varint\limits_{S^1\times S^1}  h^\ast_{12} \eta_N =\sum_{s\in h^{-1}_{12}(N)} \varint_{U_s} h^\ast_{12} \eta_N=\sum_{s\in h^{-1}_{12}(N)} \operatorname{sign}(Dh_{12}(s)),
\end{equation}

where $U_s\subset h^{-1}_{12}(D^2_\varepsilon)$ are disjoint neighborhoods of finitely many crossing points $s\in h^{-1}_{12}(N)$ such that $h_{12}|_{U_s}$ is a diffeomorphism (for $\varepsilon$ small enough), by the inverse function theorem. The last identity in the above computation is simply the
change of variables for the integral in (<ref>), and $\operatorname{sign}(Dh_{12}(s))$ depends  if $h_{12}|_{U_s}$ is orientation preserving or reversing.

A natural question which arises for $c_2(K)$, as well as other finite type invariants, is whether one may derive a combinatorial expressions in the spirit of the derivation (<ref>)–(<ref>). In the following theorem, we answer this question positively for long knots in the subspace of $\K$:

\begin{equation}\label{eq:K_N}
\K_{N}=\{K\in \K\ |\ \dot{K}(s)\nparallel N,\ s\in \R \},
\end{equation}

i.e. the space of embeddings, which are transverse to the fixed direction parallel to the vector $N$ in (<ref>). Clearly, that the path connected components of $\K_{N}$ and $\K$ are the same.

$(i)$ Given a long knot $K$ in $\K_{N}$, we have the integral invariant[for any $K\in \K_{N}$ the sequence under the limit becomes eventually constant]

\begin{equation}\label{eq:casson-integral-eta}
\begin{split}
I(K)  & = \lim_{\varepsilon\to 0}\ \Bigl(\varint\limits_{\Cnf[\R;4]} h^\ast_{13}\eta^\varepsilon_N\wedge h^\ast_{42}\eta^\varepsilon_N-\varint\limits_{\Cnf[\R,\K;3,1]} h^\ast_{1\mathbf{4}}\eta^\varepsilon_N\wedge h^\ast_{\mathbf{4}2}\eta^\varepsilon_N\wedge h^\ast_{3\mathbf{4}}\eta^\varepsilon_N\Bigr).
\end{split}
\end{equation}

$(ii)$ If $K$ admits the regular double–crossing projection onto the plane orthogonal to $N$, we obtain

\begin{equation}\label{eq:gd_c2_dbl-crossings}
I(K)=c_2(K)=\langle \vvcenteredinclude{.3}{gd_c2_chord.pdf},G_K\rangle.%=\langle \vvcenteredinclude{.3}{gd_c2_chord_unmarked.pdf},G_K\rangle.
\end{equation}

The combinatorial expressions in (<ref>)  are known as the arrow diagram formulas and were first introduced by Polyak and Viro in [28], and then further developed in subsequent works [29], [12]. We further obtain an adaptation of the arrow diagram formulas and Gauss diagrams to the setting of long knots with multiple crossings in the following

For the multiple–crossing Gauss diagram $G_K$, we have the following generalization of the formula in (<ref>)

\begin{equation}\label{eq:gd_c2_mult-crossings}
\begin{split}
I(K)=c_2(K) & =\langle \vvcenteredinclude{.3}{gd_c2_chord.pdf},G_K\rangle+ \tfrac{1}{2}\bigl(\langle \vvcenteredinclude{.3}{gd_c2_chord-left.pdf},G_K\rangle\\
&\qquad +\langle \vvcenteredinclude{.3}{gd_c2_chord-mid.pdf},G_K\rangle+\langle \vvcenteredinclude{.3}{gd_c2_chord-right.pdf},G_K\rangle\bigr).
\end{split}
\end{equation}

As a consequence of the above result we obtain a lower bound for the übercrossing number $\textit{\"u}(K)$, introduced in [3], in terms od $c_2(K)$

\[
|c_2(K)|\leq {\textit{\"u}(K) -1 \choose 3}+{ \textit{\"u}(K) -1 \choose 4}.
\]

We refer the reader to [2] for previous results concerning $\textit{\"u}(K)$.

§.§ Arrow diagrams
A general (multi-crossing) Gauss diagram of a long knot $K:\R\longrightarrow\R^3$, denoted $G_K$, is a graph (obtained from a planar projection of $K$) with:

$(i)$ a base strand - an oriented copy of $\R$, and

$(ii)$ the preimages of each crossing connected with a chord.

To incorporate the information of over and under strands of the crossing, each chord is oriented from the lower branch to the upper one.  Each chord in $G_K$ is also decorated with the crossing sign. Figure <ref> shows the Gauss diagram of $4_1$ obtained from the knot diagram shown in Figure <ref>.

A Gauss diagram representing the long figure 8 long knot.

A (long) arrow diagram $A$ is a general unsigned graph which is a copy of oriented $\R$ (base strand) with an arbitrary collection of oriented chords also called arrows, having distinct endpoints marked with $\circ$, common endpoints marked with $\bullet$ or unmarked endpoints: $\perp$. The unmarked endpoints are allowed to be either distinct or colliding. These properties are analogous to the Gauss diagrams  $G_K$,  except no signs of crossings attached to the chords.
The arrow diagrams and Gauss diagrams can be paired as follows; for  a  Gauss diagram $G=G_K$ of a long knot $K$, an embedding of an  arrow diagram $A$ in $G$ is a graph embedding of $A$ into $G$ mapping the base strand $\mathbb{R}$ of $A$ to the base strand of $G$, and chords to chords, preserving orientations and the above conventions about the markings of the endpoints: $\circ$, $\bullet$, $\perp$.  Further,  the sign of an embedding $\phi : A \longrightarrow G$  is given by[chords of $A$ will be denoted by the greek letters: $\alpha$, $\beta$,…, and chords of $G$ by lowercase letters: $g$, $h$,…]

\begin{equation}\label{eq:sign_representation}
\operatorname{sign}(\phi) = \prod_{\alpha \in A} \operatorname{sign}(\phi (\alpha)),
\end{equation}

where the $\operatorname{sign}(\phi(\alpha))$ is a sign of the arrow $g=\phi(\alpha)$ in $G$. Further,

\begin{equation}\label{eq:<A,G>}
\langle A, G\rangle = \sum_{\phi: A \rightarrow G} \operatorname{sign}(\phi),
\end{equation}

is a sum taken over all embeddings $\phi: A \longrightarrow G$ of $A$ in $G$.
A formal sum of arrow diagrams $P=\sum_i c_i A_i$ with integer coefficients is known[in the standard double crossing case] as an arrow polynomial [28], and $\langle P,G\rangle$ is defined from (<ref>) and (<ref>) by the linear extension. Also note that using the endpoints convention, we may express the arrow diagrams with unmarked endpoints in terms of the marked endpoint diagrams, for instance

\begin{equation}\label{eq:ubmarked-to-marked}
\vvcenteredinclude{.3}{gd_c2_chord_unmarked.pdf}=
\vvcenteredinclude{.3}{gd_c2_chord.pdf}+\vvcenteredinclude{.3}{gd_c2_chord-left.pdf}
\end{equation}

In a slightly different setting of signed arrow diagrams, the theorem of Goussarov [19] shows that any finite type invariant $v$ of knots can be expressed  as $\langle P_v,\,\cdot\,\rangle$ for a suitable choice of a signed arrow polynomial $P_v$. Arrow polynomials of some low degree invariants have been computed in [27, 28, 29, 35]. Apart from low degree examples, the arrow diagram formulae are known for: the coefficients of the Conway [13] and the HOMFLY-PT polynomials, [14], and the Milnor linking numbers [21, 25], see [12] for further information. All formulas obtained in these works assume that the Gauss diagrams of knots are regular double crossing diagrams.

Transforming a regular double–crossing projection into a multicrossing projection. All unmarked chords have a positive sign. The chords marked in red correspond to chords that contribute to the $c_2$ count.  Note that in figure $(f)$, the triple crossing between the $1^{st}, 3^{rd}$, and $4^{th}$ strand creates two arrow subdiagrams which cancel each other out, as is depicted in row 2 of Table <ref>.

Long Knot Diagram     Long Gauss diagram
(a)     .7     .7

(b)     .7     .7

(c)      .7     .7

(d)        .7     .7

(e)        .7     .7

(f)          .7     .7

Table <ref> shows various projections of the long trefoil knot and the corresponding multiple–crossing Gauss diagrams: $G_a$, $G_b$, $G_c$, $G_d$, $G_e$, and $G_f$, which represent moves transforming a regular double–crossing projection into a multi-crossing projection.  Using the above definitions, we may calculate various pairings of arrow diagrams and Gauss diagrams. Since $G_a$ is a regular double crossing diagram (<ref>) yields

\[
c(4_1)=\langle \vvcenteredinclude{.3}{gd_c2_chord.pdf},G_a\rangle=\langle \vvcenteredinclude{.3}{gd_c2_chord_unmarked.pdf},G_a\rangle=1.
\]

However, the formula in  (<ref>) does not extend to the multi-crossing Gauss diagrams, for instance

\[
\begin{split}
\langle \vvcenteredinclude{.3}{gd_c2_chord.pdf},G_c\rangle & =0,\quad \langle \vvcenteredinclude{.3}{gd_c2_chord_unmarked.pdf},G_c\rangle=2,\\
\langle \vvcenteredinclude{.3}{gd_c2_chord.pdf},G_f\rangle & =0,\quad \langle \vvcenteredinclude{.3}{gd_c2_chord_unmarked.pdf},G_f\rangle=2.
\end{split}
\]

Formula (<ref>) yields (for the multi-crossing Gauss diagram $G_e$)

\[
\begin{split}
c_2(K) & =\langle \vvcenteredinclude{.3}{gd_c2_chord.pdf},G_f\rangle+ \tfrac{1}{2}\bigl(\langle \vvcenteredinclude{.3}{gd_c2_chord-left.pdf},G_f\rangle +\langle \vvcenteredinclude{.3}{gd_c2_chord-mid.pdf},G_f\rangle\\
& +\langle \vvcenteredinclude{.3}{gd_c2_chord-right.pdf},G_f\rangle\bigr) = 0+\tfrac{1}{2}\bigl(1+(1-1)+1\bigr)=1.
\end{split}
\]

§.§ Connections to existing work
In [29], Polyak and Viro
equate the combinatorial formula (<ref>), first proposed in [28], with a degree $\deg(h)$ of the map $h=h_{13}\times h_{42}\times\operatorname{id}$. In order to properly define this degree they extend the compactified domain[an $S^2$ factor is needed to make the boundary of $\Cnf[I;4]\times S^2$ compatible with the boundary of $\Cnf[I,\K_{S^1};3,1]$] $\Cnf[I;4]\times S^2$ of $h$ in (<ref>) by gluing $\Cnf[I,\K_{S^1};3,1]$ along the principal faces, and identifying hidden faces via Kontsevitch's involutions to obtain a $6$ dimensional cell complex $\mathcal{C}$.
Since $\mathcal{C}$ has a well defined fundamental class, $\deg(h)$ can be defined homologically as the pullback of the fundametal class of the codomain $(S^2)^3$

\begin{equation}\label{eq:c_2=deg(h)}
c_2(K)=\deg(h)=\langle h_\ast([\mathcal{C}]),[(S^2)^3]\rangle.
\end{equation}

Whether we can perform this computation by integrating a differential form is a different issue because $\mathcal{C}$ is not a closed manifold. In [8], Bott and Taubes use the symmetric Gauss form $\omega$ on each $S^2$ factor to obtain the formula (<ref>), we review the details of this calculation for long knots in Section <ref> and formula (<ref>).
If $K$ has a regular double crossing projection (onto a plane orthogonal to $N$), $(N,N,N)$ is a regular value of $h$ and the local degree formula yields the arrow diagram count[for sufficiently small $\varepsilon$]: $c_2(K)=\langle \vvcenteredinclude{.25}{gd_c2_chord.pdf},G_K\rangle$, which we also obtain from
our integral (<ref>) in Section <ref> by showing that the contribution of the second term in (<ref>) vanishes.

In contrast to the above, “cut and paste” approach, we construct the localized integral (<ref>) directly using the Bott–Taubes integration technique with the form $\eta_N$ (<ref>). Since $\eta_N$ is not antisymmetric, we cannot use the Kontsevitch's involutions for proving vanishing over the hidden and anomalous faces. As a consequence, we obtain the invariant defined over a subspace $\K_{N}$, (<ref>) of $\K$, which has the same path connected components as $\K$, thus we do not loose any information. Further, note that the local degree argument generally fails for knots which do not have regular double crossing projection in the direction of $N$, thus $\varint\limits_{\Cnf(I;4)} h^\ast_{13}\eta_N\wedge h^\ast_{42}\eta_N$ will not give a correct value for $c_2(K)$,
while the formula (<ref>) works for any knot in $\K_{N}$, in particular for the multicrossing knot diagrams yields (<ref>).

The techniques outlined above can be regarded as a discretization of the Bott and Taubes  configuration space integral (<ref>).
In [26], Lin and Wang approached this discretization in a different way, rather than replacing the from $\omega$ with $\eta_N$ they adjusted the parametrization of the knot $K$ to lie almost entirely in the plane orthogonal to $N$ except $\varepsilon$–small overpasses (assuming $K$ admits a regular projection onto that plane), denoted by $K_\varepsilon$. Then, they examined the limit of the integral (<ref>) over $K_\varepsilon$ as $\varepsilon\to 0$. Although technically similar, this limit does not yield the local degree expression
in (<ref>). In particular the second term $\varint\limits_{\Cnf(I,K;3,1)} \omega_{1\mathbf{4}}\wedge\omega_{2\mathbf{4}}\wedge\omega_{3\mathbf{4}}$ (the “tripod” integral)
in (<ref>) does not vanish in the limit.  The first term $\frac{1}{4}\varint\limits_{\Cnf(I;4)} \omega_{13}\wedge\omega_{24}$ gives a signed crossing count in the limit, reminiscent[where the all arrow directions are allowed in the arrow diagram] of (<ref>), plus a term proportional to the crossing number $\operatorname{Cr}(K)$ of $K$. We expect that computations of Lin and Wang, can be also obtained in our setting by using the averaged from $\eta_{Av}=\tfrac{1}{2}(\eta_N+\eta_S)$ in place of $\omega$. One advantage of $\eta_{Av}$ is that it is antisymmetric, thus we could use the involutions of [22] to show vanishing over certain hidden faces, and apply our technique (Section <ref>) to show vanishing over the anomalous face. On the other hand, nonvanishing of the tripod integral term in the limit prevents us from a straightforwad identification of the invariant with the general arrow diagram formula in (<ref>).  One way to address this is to consider the more general setting of tinkertoy diagrams of Thurston [32], this however results in counting features[such as similar triangles in the knot projection] of the knot projection not tied only to the signed crossings, see e.g. [29].

In summary, the techniques investigated in this work, and presented in the case of $c_2(K)$,  show that some complications of the original Bott–Taubes integration procedure (such as vanishing over the anomalous face) can be simplifed when working with localized forms, which may provide a correct setting for the diagram cochain complex for knots  (c.f. [11], [24]) in the classical dimension $n=3$.
It may also be a proper setting to prove the extension of the Goussarov's result in [19], to unsigned arrow diagrams in (<ref>). We plan to invesigate these questions in the future work.

§.§ Organization of the paper
The paper is organized as follows. In Section <ref> we review the Bott–Taubes technique and formulate the integral (<ref>) in terms of the pushforward operator. The proof of Theorem <ref> is given in Section <ref>, where in the first part we follow the Bott–Taubes technique to prove invariance of the integral (<ref>) and in the second part we show (<ref>) via the local degree formula in the style of [9]. Theorem <ref> is proven, via combinatorial techniques, in Section <ref>. Finally, in Section <ref>, we focus on pedal projections and obtain Corollary <ref> as a consequence.

§.§ Acknowledgments
This paper builds on the results of the first author's doctoral thesis [10]. The second author acknowledges the partial support by Louisiana Board of Regents
Targeted Enhancement Grant 090ENH-21.

§ REVIEW OF THE BOTT–TAUBES INTEGRATION

§.§ Preliminaries on the Stokes theorem for fiber bundles and pushforwards
Consider a smooth fiber bundle $\pi:E\longrightarrow B$ of manifolds $E$ and $B$ of dimensions $m+n$ and $m$, with fiber $F$ compact manifold with boundary of dimension $n$. Let us review the definition of the pushforward of differential forms  in the style of [9],

\begin{equation}\label{eq:pi_ast}
\pi_\ast:\Omega^\ast(E)\longrightarrow \Omega^{\ast-n}(B),\quad \text{also denoted by}\quad \varint_{F}=\pi_\ast.
\end{equation}

It suffices to define $\pi_\ast$ in a local trivialization $U\times W$ of $E$ where $U$ is a neighborhood in $B$, and $W=[0,\varepsilon)\times V\subset F$ is intersecting the boundary or $W=(0,\varepsilon)\times V$ is in the interior of $F$.
Let $(\mathbf{x},\mathbf{t})=(x_1,\ldots,x_m,t_1,\ldots, t_n)$ be coordinates on $U\times W$ (where either $t_1\geq 0$ or $t_1>0$); a differential form $\beta\in \Omega^{k}(E)$ can be locally expressed as a sum of two types of terms, the type (I) forms which do not contain $d\mathbf{t}=dt_1\wedge\ldots\wedge dt_n$ and type (II) which contain it. The map $\pi_\ast$ is defined by

\begin{equation}\label{eq:pi_ast-def}
\pi_\ast \beta=\begin{cases}
\text{(I)}\ & \pi^\ast \phi \wedge f(\mathbf{x},\mathbf{t})\, dt_{i_1}\wedge dt_{i_2}\wedge\ldots\wedge dt_{i_r}  \longrightarrow 0, \quad i_1<i_2<\ldots<i_r,\ r<n; \\
\text{(II)}\ & \pi^\ast \phi \wedge f(\mathbf{x},\mathbf{t})\, d\mathbf{t}  \longrightarrow \phi \wedge \bigl(\varint_W f(\mathbf{x},\mathbf{t})\bigr)\, d\mathbf{t},
\end{cases}
\end{equation}

where $\phi$ is a compactly supported form on $U\subset B$, $f$ has a compact support on $W$, and $d\mathbf{t}=dt_{1}\wedge dt_{2}\wedge\ldots\wedge dt_{n}$.
We have the following version of Stokes' theorem for fiber bundles,

\begin{equation}\label{eq:stokes-fiber}
d \pi_\ast\beta = \pi_\ast d\beta + (-1)^{\operatorname{deg}(\phi)}(\partial \pi)_\ast \beta,\quad \operatorname{deg}(\phi)=k-n+1,
\end{equation}

where $(\partial \pi)_\ast \beta$ will be called the boundary pushforward.
For a type (II) form $\beta$, just as in  <cit.>, we have

\[
d\pi_\ast \beta = \pi_\ast d\beta.
\]

By (<ref>), for a type (I) form $\beta=\pi^\ast\phi \wedge f(\mathbf{x},\mathbf{t})\, dt_{i_1}\wedge dt_{i_2}\wedge\ldots\wedge dt_{i_r}$, $\pi_\ast \beta=0$ and thus $d\pi_\ast \beta=0$. Further,

\[
\begin{split}
\pi_\ast d\beta & =\pi_\ast\bigl((d\pi^\ast\phi) \wedge f(\mathbf{x},\mathbf{t})\, dt_{i_1}\wedge dt_{i_2}\wedge\ldots\wedge dt_{i_r}\bigr)\\
& +(-1)^{\operatorname{deg}(\phi)}\sum_j \pi_\ast\bigl(\pi^\ast\phi \wedge \frac{\partial f}{\partial t_j}(\mathbf{x},\mathbf{t}) dt_j\wedge dt_{i_1}\wedge dt_{i_2}\wedge\ldots\wedge dt_{i_r}\bigr),
\end{split}
\]

where the first term vanishes by (<ref>), the second term is nonzero only for $r=n-1$, and can be written as

\[
\pi_\ast d\beta =(-1)^{\operatorname{deg}(\phi)}\sum^n_{j=1} \phi \wedge \bigl(\varint_{W}\frac{\partial f}{\partial t_j}(\mathbf{x},\mathbf{t}) dt_j\wedge dt_{1}\wedge \ldots\wedge \widehat{dt_{j}}\wedge\ldots\wedge dt_{n}\bigr).
\]

From the definition of $W$ we can show (similarly as in <cit.>) the right hand side vanishes for
$W=(0,\varepsilon)\times V$, (since $\beta$ is compactly supported) and each term in the sum is zero for $j>1$, which leaves only $j=1$ and $W=[0,\varepsilon)\times V$, ($0\leq t_1<\varepsilon$) a possible nonzero contribution. Indeed, taking into account the orientation[consistent with the outer normal convention] of $\partial W=\{0\}\times V$, by $-dt_2\wedge\ldots\wedge dt_{n}$

\[
\begin{split}
\pi_\ast d\beta & =(-1)^{\operatorname{deg}(\phi)} \phi \wedge \bigl(\varint_{[0,\varepsilon)\times V}\frac{\partial f}{\partial t_1}(\mathbf{x},\mathbf{t}) dt_{1}\wedge \ldots\wedge dt_{n}\bigr)\\
& = (-1)^{\operatorname{deg}(\phi)} \phi \wedge \varint_{V} \bigl(\bigl(\lim_{t_1\to \varepsilon} f(\mathbf{x},\mathbf{t})\bigr)- f(\mathbf{x},\mathbf{t})\bigl|_{t_1=0}\bigr) dt_2\wedge\ldots\wedge dt_{n}
\end{split}
\]

\[
= -(-1)^{\operatorname{deg}(\phi)} \phi \wedge \varint_{\{0\}\times V} f(\mathbf{x},\mathbf{t}) dt_2\wedge\ldots\wedge dt_{n}=(-1)^{\operatorname{deg}(\phi)} \phi \wedge \varint_{\partial W} f(\mathbf{x},\mathbf{t}) dt_2\wedge\ldots\wedge dt_{n},
\]

(since $f$ is compactly supported on $W$ the limit of $\varepsilon$ at $0$ vanishes) and since $\operatorname{deg}(\phi)=\operatorname{deg}(\beta)-n$. Therefore, we obtain the formula (<ref>) in the Stokes' theorem, if $(\partial \pi_\ast)$ is defined on type (I) and (II) forms as follows $(\partial \pi_\ast)\beta=$

\begin{equation}\label{eq:partial-pi_ast-def}
\begin{cases}
\text{(I)} &   \pi^\ast \phi \wedge f(\mathbf{x},\mathbf{t})\, dt_{i_1}\wedge dt_{i_2}\wedge\ldots\wedge dt_{i_r}  \longrightarrow 0,\ r<n-1 \textrm{ or } i_j=1 \textrm{ for some } j; \\
&   \pi^\ast \phi \wedge f(\mathbf{x},\mathbf{t})\, dt_{2}\wedge dt_{3}\wedge\ldots\wedge dt_{n}  \longrightarrow \phi\wedge \varint_{\partial W} f(\mathbf{x},\mathbf{t}) dt_2\wedge\ldots\wedge dt_{n}; \\
\text{(II)} 	&  \pi^\ast \phi \wedge f(\mathbf{x},\mathbf{t})\, dt_{1}\wedge dt_{2}\wedge\ldots\wedge dt_{n} \longrightarrow 0,
\end{cases}
\end{equation}

where $\partial W\neq \varnothing$, just for $W=[0,\varepsilon)\times V$, and $\partial W=\{t_1=0\}=\{0\}\times V$. In the calculations we encounter here, the boundary will be partiotioned into pieces which possibly overlap as sets of zero measure, i.e. $\partial W=Q_1\cup\ldots\cup Q_m$, therefore restricting the integral in (<ref>)
to $Q_i$ yields $(\partial \pi_\ast)\beta\bigl|_{Q_i}$ and

\begin{equation}\label{eq:bdry-pushforward-faces}
(\partial \pi_\ast)\beta=\sum^m_{i=1} (\partial \pi_\ast)\beta\bigl|_{Q_i}.
\end{equation}

We have the following basic result about boundary pushforwards.

Suppose $h,h':E,E'\longrightarrow M$ are smooth maps from the total spaces $E$, $E'$  of the bundles $\pi,\pi':E^{n+m}, (E')^{n+m}\longrightarrow B^m$ where a manifold $M$ is of dimension $m+n$. Assuming that the fibers $F$ and $F'$ of $E$ and $E'$ have a common boundary $\partial F'=\partial F$ and $h\bigl|_{\partial F}=h'\bigl|_{\partial F}$ for $\beta\in \Omega^k(M)$, we have

\begin{equation}\label{eq:partial-pi-ast-equal}
(\partial \pi_\ast)h^\ast\beta=\pm(\partial \pi_\ast)h'^\ast\beta.
\end{equation}

where the sign is taking into account the orientation.

Consider a local trivialization $U\times W\subset E$, $W=[0,\varepsilon)\times V$ with coordinates $(\mathbf{x},\mathbf{t})\in U\times W$, and $U'\times W'\subset E'$, $W'=[0,\varepsilon)\times V'$   with coordinates $(\mathbf{x}',\mathbf{t}')\in U'\times W'$.
In the notation from
the above paragraph, and let $\mathbf{z}=(z_1,\dots,z_{n+m})$ be local coordinates for $M$ around a point $\mathbf{z}_0=h(\mathbf{x}_0,\mathbf{t}_0)=h'(\mathbf{x}'_0,\mathbf{t}'_0)$, $(\mathbf{x}_0,\mathbf{t}_0)\in \partial F$, $(\mathbf{x}'_0,\mathbf{t}'_0)\in \partial F'$. In these local coordinates

\[
\begin{split}
z_j & =h_j(\mathbf{x},\mathbf{t}),\qquad z_j=h'_j(\mathbf{x}',\mathbf{t}')\qquad j=1\ldots n+m,\\
dz_j & =\sum_s \tfrac{\partial}{\partial x_s} h_j(\mathbf{x},\mathbf{t})\, dx_s+\sum_r \tfrac{\partial}{\partial t_r} h_j(\mathbf{x},\mathbf{t})\, dt_r,\\
dz_j & =\sum_s \tfrac{\partial}{\partial x'_s} h'_j(\mathbf{x}',\mathbf{t}')\, dx'_s+\sum_r \tfrac{\partial}{\partial t'_r} h'_j(\mathbf{x}',\mathbf{t}')\, dt'_r
\end{split}
\]

Given $\beta=\displaystyle\sum_I f_I(\mathbf{z}) dz_I$, where $I=\{i_1,\dots,i_k\}\subset\{1,\dots,n+m\},$ and $|I|=k$, $dz_I=dz_{i_1}\wedge \ldots\wedge dz_{i_k}$ the pull–back of $\beta$ under $h$ (and analogously under $h'$) to $U\times W$ is a sum of type (I) and (II) terms; however, because of the definition of $\partial \pi_\ast$, we only consider the relevant type (I) terms:

\[
h^\ast(f_I(\mathbf{z}) dz_I) = (f_I\circ h)(\mathbf{x},\mathbf{t}) H_J(\mathbf{x},\mathbf{t}) dx_J\wedge dt_{2}\wedge dt_{3}\wedge\ldots\wedge dt_{n}
\]

\[
(h')^\ast(f_I(\mathbf{z}) dz_I) = (f_I\circ h')(\mathbf{x}',\mathbf{t}') H'_J(\mathbf{x}',\mathbf{t}') dx'_J\wedge dt'_{2}\wedge dt'_{3}\wedge\ldots\wedge dt'_{n},
\]

where $|J|=k-n+1$, and $H_J(\mathbf{x},\mathbf{t})$, $H'_J(\mathbf{x}',\mathbf{t}')$ are polynomials in partial derivatives:
$\tfrac{\partial}{\partial x_s} h_j$,  $\tfrac{\partial}{\partial t_r} h_j$ with $r\neq1$, and $\tfrac{\partial}{\partial x'_s} h'_j$,  $\tfrac{\partial}{\partial t'_r} h'_j$ respectively.
Under the assumption $h\bigl|_{\partial F}=h'\bigl|_{\partial F}$, we have that

\[(f_I\circ h)(\mathbf{x},\mathbf{t})\big|_{t_1=0}=(f_I\circ h')(\mathbf{x}',\mathbf{t}')\big|_{t'_1=0}.\]

Moreover, the partial derivatives of $h$ and $h'$ with respect to $x_s$, $t_r$ and $x'_s$, $t'_r$ for $r\neq1$ must also agree when restricted to $\{t_1=0\}$.  Therefore, $H_J(\mathbf{x},\mathbf{t})\big|_{t_1=0} =(-1)^{\operatorname{sign}(\partial h')} H'_J(\mathbf{x}',\mathbf{t}')\big|_{t'_1=0}$ where the sign accounts for the boundary orientation. Thus the equality (<ref>) follows from the definition in (<ref>).

§.§ Integrating over the compactified domains

In this section we show, in the style of [8, 33], how to express the integrals (<ref>), (<ref>)  in terms of the pushforward operator $\pi_\ast$.
Recall that we work with the subspace of long knots  $\K_N$ defined in (<ref>) and consider the trivial fiber bundle $\Cnf[\R;4]\times\K_N$ with projection map

\begin{equation}\label{eq:pi-projection-C_4}
\pi:\Cnf[\R;4]\times \K_N\longrightarrow \K_N
\end{equation}

and fiber $\pi^{-1}(K)=\Cnf[\R;4]$ for $K\in\K_N$.
Define an evaluation map

\begin{equation}\label{eq:ev_4}
\operatorname{ev}_4:  \Cnf[\R;4]\times \K_N\longrightarrow \Cnf[\R^3;4]
\end{equation}

as an extension of the map

\[
((x_1,x_2,x_3,x_4),K)\longrightarrow (K(x_1),K(x_2),K(x_3),K(x_4)),
\]

from the open configuration spaces.

Chord diagram associated to the map $h_4$.

Also, let $h_{ij} :\Cnf[\R^3;4]\longrightarrow S^2$ be the smooth extension of the standard Gauss map to the associated compactifications

\begin{equation}\label{eq:h_ij}
h_{ij} :\Cnf(\R^3;4)\longrightarrow S^2,\qquad
(\bx_1,\bx_2,\bx_3,\bx_4)\longrightarrow \frac{\bx_j-\bx_i}{|\bx_j-
\bx_i|}.
\end{equation}

Such extensions are uniquely defined as shown in [31].
For the $2$–form $\eta_N$, given in (<ref>), $\eta_{N}\times\eta_{N}=p^\ast_1\eta_{N}\wedge p^\ast_2\eta_{N}$ is a $4$–form over $S^2\times S^2$. Let

\begin{equation}\label{eq:beta}
\beta=h^\ast_4(\eta_{N}\times \eta_{N}),\quad h_4=(h_{13}\times h_{42})\circ \ev_4,
\end{equation}

be the pullback form over $\Cnf[\R;4]\times\K_N$. Note that the map $h_4$ in (<ref>) can be formally encoded by the familiar arrow diagram
$\vvcenteredinclude{.3}{gd_c2_chord_unmarked.pdf}$ of Section <ref>. The encoding is shown on Figure <ref>, where the coordinates on $\Cnf[\R;4]$ and the Gauss maps are indicated.

By restricting the map $\pi$ defined in (<ref>) to $\Cnf[\R;4]\times\K_N$, the pushforward of $\beta$ yields a function on $\K_N$ (i.e. the degree $0$–form on $\K_N$)

\begin{equation}\label{eq:pi-beta}
\pi_\ast\beta(K) = \varint_{\Cnf[\R;4]} h^\ast_4(\eta_{N}\times \eta_N),\qquad K\in \K_N,
\end{equation}

which is understood in the sense of Appendix <ref>: The function $\pi_\ast\beta$ is defined over plots $\mathcal{K}:U\longrightarrow \K_N$, where $\pi_\ast\beta=\{(\pi_\ast\beta)_{\mathcal{K}}\}_{\mathcal{K}:U\longrightarrow \K_N}$, and $(\pi_\ast\beta)_\mathcal{K}$ is given by the pushforward  of $(\beta)_\mathcal{K}$,in (<ref>), over the trivial projection $\Cnf[\R;4]\times U\longrightarrow U$.
The question of whether this function is a knot invariant is a question of whether it is locally constant;  which  is true if $d \pi_\ast\beta =0$ (i.e. the $0$–form $\pi_\ast\beta$ is closed). Using (<ref>) and $d\beta=0$, we obtain

\begin{equation}\label{eq:stokes-fiber2}
d\pi_\ast\beta=\pi_\ast d\beta - (\partial \pi)_\ast \beta=-(\partial \pi)_\ast \beta,
\end{equation}

where $(\partial \pi)_\ast \beta$ is the sum of pushforwards along the codimension one faces of $\Cnf[\R;4]$ as defined in (<ref>).

Unfortunately, the $1$–form on right hand side of (<ref>) vanishes for some, but not all, of the boundary faces of $\Cnf[\R;4]$; therefore a “correction term” is needed to obtain a knot invariant from $\pi_\ast\beta$ in (<ref>). This correction term is constructed as an integral over a connected component of $\Cnf[\K_N,\R^3;3,1]$, which is also a manifold with corners, given as a pullback bundle over $p$ in the following diagram

\[
\xymatrix{
\Cnf[\K_N,\R^3;3,1] \ar[rr] \ar^-{\ev_{3;1}}[rr] \ar@{->}_-{}[d]  & & \Cnf[\R^3;4] \ar^-{p}[d] \\
\Cnf[\R;3]\times \K_N  \ar^-{\ev_3}_-{\ }[rr] & &  \Cnf[\R^3;3].
\]

The projection $p$ is the extension of the projection from $\Cnf(\R^3;4)$ to $\Cnf(\R^3;3)$ given by skipping the last coordinate. By definition

\begin{equation*}\label{eq:Cnf_K}
\Cnf[\K_N,\R^3;3,1]=\{(x, K;\mathbf{x})\in (\Cnf[\R;3]\times \K_N)\times \Cnf[\R^3;4]\ |\ \ev_3(x, K)= p(\mathbf{x})\},
\end{equation*}

so that a fiber of $\Cnf[\K_N,\R^3;3,1]$ over $K\in \K_N$ is the subset of $\Cnf[\R^3;4]$ restricted to configurations $(\mathbf{x}_1,\mathbf{x}_2, \mathbf{x}_3, \mathbf{x}_4)$, where $\mathbf{x}_1,\mathbf{x}_2, \mathbf{x}_3$ “sit” on the knot $K$.  We refer to $\mathbf{x}_4$ as the point “off the knot" and $x=(x_1,x_2,x_3)\in\Cnf[\R;3]$.  The top row evaluation map in the above diagram is given by

\[
\operatorname{ev}_{3;1}: \Cnf[\K_N,\R^3;3,1]\subset\Cnf[\R;3]\times \K_N\times \Cnf[\R^3;4] \longrightarrow \Cnf[\R^3;4],
\]

which is the extension of

\[
((x_1,x_2,x_3),K,(\mathbf{x}_1,\mathbf{x}_2, \mathbf{x}_3, \mathbf{x}_4))\longrightarrow (K(x_1),K(x_2),K(x_3),\mathbf{x}_4),
\]

the map $\ev_3$ is analogous to $\ev_4$ in (<ref>).
It is then clear that a fiber of the obvious projection

\[
\pi':\Cnf[\K_N,\R^3;3,1]\longrightarrow \K_N
\]

is $6$ dimensional.
Let us define

\begin{equation}\label{eq:beta'-h'}
\begin{split}
\beta' & =h^\ast_{3;1}(\eta_{N}\times \eta_N\times \eta_{N}),\qquad \text{where}\\
h_{3;1} & :\Cnf[\R;3]\times \K_N\times \Cnf[\R^3;4]\longrightarrow S^2\times S^2\times S^2,\\
h_{3;1} & =(h_{1\mathbf{4}}\times h_{\mathbf{4}2}\times h_{3\mathbf{4}})\circ \ev_{3;1},
\end{split}
\end{equation}

this map is encoded by the tripod diagram of Figure <ref>.

Trivalent tripod diagram associated to $h_{3;1}$.

The required correction term for $\pi_\ast\beta$ is defined (over the plots of $\K_N$) as the pushforward $\pi'_\ast \beta'=\{(\pi'_\ast \beta')_{\mathcal{K}}\}_{\mathcal{K}:U\longrightarrow \K_N}$:

\[
\pi'_\ast \beta'=\varint_{\Cnf[\K_N,\R^3;3,1]} h^\ast_{3;1}(\eta_{N}\times \eta_N\times \eta_{N}).
\]

We may now rewrite the integral formula (<ref>) of Theorem <ref> as

\begin{equation}\label{eq:casson-integral-eta-pi_ast}
\end{equation}

Clearly, it requires a proof
that $\pi'_\ast\beta'$ provides a required correction for $\pi_\ast\beta$.
Such proof, as well as the  approach detailed above, was first provided for closed knots $\K_{S^1}$ in the work of Bott and Taubes in [8], with the standard rotationally invariant volume form $\omega$ on $S^2$ in place of $\eta_N$. The main result of [8] yields the integral stated in (<ref>) over the compactified domains. Using the
pushforwards (<ref>) can be stated as[Note the rational coefficients, which appear due to a different count of principal faces.]

\[
c_2(K)=\frac{1}{4}\pi_\ast \alpha(K)-\frac{1}{3}\pi'_\ast\alpha'(K),
\]

where $\alpha=h^\ast_4(\omega\times\omega)$ and $\alpha'=h^\ast_{3;1}(\omega\times\omega\times\omega)$.
The work in [8] was further extended in [32, 11, 33, 24]. In [11] and [24] the authors construct a diagram cochain complex which puts the correspondence between Figures <ref>, <ref> and maps: $h_4$, $h_{3;1}$ in the general setting of cohomology of long knots and links.

§ PROOF OF THEOREM <REF>

Suppose that $K_0$ and $K_1$ are two isotopic knots in $\K_N$ defined in (<ref>); with regular projections onto the plane orthogonal to some fixed direction $N$.  Since any two isotopic knots are related by a sequence of the Reidemeister moves ([5], [30]), without loss of generality we could  assume that $K_0$ and $K_1$ are connected by a regular isotopy, however for our purposes we only require a smooth path in $\K_N$, i.e.

\begin{equation}\label{eq:smooth-isotopy}
K:\R\times [0,1] \longrightarrow \R^3,\qquad (\,\cdot\,,z)\longrightarrow K(\,\cdot\,,z)\in \K_N.
\end{equation}

Specifically, $K$ satisfies the following

* $K(\,\cdot\,,0)=K_0$, $K(\,\cdot\,,1)=K_1$,

* The unit vector[here $\dot{K}=\frac{\partial}{\partial x} K$ denotes a derivative along the knot]  $\frac{\dot{K}(x,z)}{|\dot{K}(x,z)|}$ never points along $N$, i.e.  there exists a $\delta$–disk: $D^2_\delta(N)$ around $N\in S^2$, such that $\frac{\dot{K}(x,z)}{|\dot{K}(x,z)|}\notin D^2_\delta(N)\subset S^2$ for all $x$ and $z$. Generally, we will require that $\varepsilon<\delta$ in the definition of $\eta_N$ in (<ref>).

* Since $K(\,\cdot\,,z)$ is a long knot in $\K_N$, and each $z\in [0,1]$, and $x\not\in [-1,1]$, we have
$\dot{K}(x,z)=X$, where $X$ is a fixed vector parallel to the line perpendicular to $N$ (we may choose the $x$ axis for that purpose).

With $I(K_i)$, $i=0,1$ defined in (<ref>) or (<ref>) we need to show that $I(K_0)=I(K_1)$. Observe that $I(K(\,\cdot\, ,z))$ is a function of $z\in [0,1]$, and

\[
I(K_1)-I(K_0)=\varint^1_0 \frac{\partial}{\partial z} I(K(\,\cdot\, ,z))\, dz =\varint^1_0 d I(K(\,\cdot\, ,z)).
\]

Therefore, it suffices to verify

\begin{equation}\label{eq:dI=0}
d I(K(\,\cdot\, ,z))=0.
\end{equation}

Considering the path (<ref>), as a plot in the knot space $\K_N$:

\begin{equation}\label{eq:plot-K}
\mathcal{K}:[0,1]\longrightarrow \K_N,\quad \mathcal{K}(z)=K(\,\cdot\,,z),
\end{equation}

in the differential form notation of Appendix <ref>, we obtain the $1$–form

\begin{equation}\label{eq:dI}
d I(K(\,\cdot\, ,z)) =d \bigl((\pi_\ast\beta)_{\mathcal{K}} -(\pi'_\ast\beta')_{\mathcal{K}}\bigr)
=  -((\partial \pi)_\ast \beta)_{\mathcal{K}} +((\partial \pi')_\ast \beta')_{\mathcal{K}},
\end{equation}

where the second identity follows from (<ref>).
Recall from (<ref>), $(\partial \pi)_\ast \beta$ and $\partial \pi')_\ast \beta'$ can be written as the sum over the boundary faces (or strata) of $\Cnf[\R;4]$, and $\Cnf[\K_N,\R^3;3,1]$ respectively. In order to compute the sums it is sufficient to compute in local coordinates near the boundary faces of $\Cnf[\R;4]$ and $\Cnf[\K_N,\R^3;3,1]$ respectively.

We will follow the description of local coordinates near the boundary faces given in [8, 33], for equivalent approaches one may refer to  [11] or [31].
Specifically, every face (stratum) of[Similarly for $\Cnf[\K, M;n,m]$.] $\Cnf[M;n]$ denoted by $S_{\mathcal{A}}$, can be labeled by $|\mathcal{A}|=j$ nested subsets $\mathcal{A}=\{A_1,\ldots, A_j\}$, where  $A_k\subset\{1,2,\ldots,n\}$, $|A_k|\geq 2$ and for $A_p, A_q\in \mathcal{A}$:

\[
A_p\cap A_q=\varnothing,\quad \text{or}\quad A_p\subseteq A_q.
\]

As a direct consequence of coordinates description on $\Cnf[M;n]$, one obtains that the codimension of $S_{\mathcal{A}}$ is $j$.  In particular, faces of $\Cnf[M;n]$ of codimension $1$ correspond to a subset of the $n$ points either all colliding at the same time, or escaping to infinity at the same time. In Appendix <ref>, we provide a more thorough summary of these constructions.

§.§ Faces of $\partial \Cnf[\R;4]$ appearing in $((\partial \pi)_\ast \beta)_{\mathcal{K}}$
Let us first consider $\Cnf[\R;4]$, the above description yields the following the codimension one faces of $\Cnf[\R;4]$ indexed by the following subsets of $\{1,2,3,4,\pm \infty\}$:

\begin{equation}\label{eq:Cnf[R;4]-faces}
\begin{split}
\textit{ Principal Faces: }&\{1,2\},\{2,3\},\{3,4\}\\
\textit{ Hidden Faces: }&\{1,2,3\},\{2,3,4\}\\
\textit{ Anomalous Face: }&\{1,2,3,4\}\\
\textit{ Faces at Infinity: }&\{1\to-\infty\},\{1,2\to-\infty\},\{1,2,3\to-\infty\},\{1,2,3,4\to-\infty\},\\
&\{4\to+\infty\},\{3,4\to+\infty\}, \{2,3,4\to+\infty\},\{1,2,3,4\to+\infty\}.\\
\end{split}
\end{equation}

The pushforward $((\partial \pi)_\ast \beta)_{\mathcal{K}}$ will not vanish on the principal faces, i.e. faces indexed by $\{1,2\}$, $\{2,3\}$, $\{3,4\}$;  this non-vanishing is what requires the addition of the correction term $\pi_\ast\beta'$ in (<ref>).  On all other faces, the pushforward of the restriction of $\beta$ to that face is zero. The vanishing of the pushforwards of $\beta$ and $\beta'$ on non-principal faces will be shown in further sections.  First consider $A$ where $1\in A$; then $S_A$ is the face where the first $a$ points on the knot collide away from $\infty$ in $\Cnf[\R;n]$.  Local coordinates for an open neighborhood of $S_A=\{r=0\}$ are given by

\begin{equation}\label{eq:phi_A}
\begin{split}
&\phi_A:\Cnf(\R;a)\times\Cnf(\R;n-a+1)\times[0,\infty)\times [0,1] \longrightarrow\Cnf[\R;n]\times [0,1]\\
&(u_1,\dots,u_a,x_1,\dots,x_{n-a+1},r, z)\longrightarrow  (x_1+ru_1,\dots,x_1+ru_a, x_2,\dots, x_{n-a+1}, z),
\end{split}
\end{equation}

such that

\[
(1)\  x_i\neq x_j,\   i\neq j,\qquad
(2)\ \displaystyle\sum_{i=1}^a u_i=0,\qquad
(3)\ \displaystyle\sum_{i=1}^a |u_i|^2=1.
\]

With the help of the above coordinates, the pushforwards of $\beta$ along each bundary face $S_A$ to the above coordinates by

\begin{equation}\label{eq:partial-beta-restr}
(\partial \pi\bigl|_{S_A})_\ast \beta=(\partial \pi\bigl|_{r=0})_\ast (\phi^\ast_A\beta).
\end{equation}

Clearly, we have $(\partial \pi)_\ast \beta =\sum_A (\partial \pi\bigl|_{S_A})_\ast \beta$.

§.§ Computations for $((\partial \pi)_\ast \beta)_{\mathcal{K}}$ along the principal faces
For the principal faces: $A=\{1,2\}$, $\{2,3\}$, $\{3,4\}$, we may simplify the above coordinates. Additionally, for the purpose of computing the pushforwards $((\partial \pi\bigl|_{S_{i,j}})_\ast \beta)_{\mathcal{K}}$, we
add $[0,1]$ (i.e. the domain of the plot $\mathcal{K}:[0,1]\longrightarrow \K_N$, (<ref>)) and additional “dummy” $S^2$ factor
as follows[it is $\,\cdot\,+r$ (not $\,\cdot\,-r$) since $x_1<x_2<x_3<x_4$.]

\[
\hat{\phi}_{i,j}:\Cnf(\R;3)\times[0,\infty)\times S^2\times [0,1]  \longrightarrow\Cnf[\R;4]\times S^2\times [0,1],
\]

\begin{equation}\label{eq:phi_ij}
\begin{split}
\hat{\phi}_{1,2}:(x_1,x_2,x_3,r,\xi,z) & \longrightarrow  (x_1,x_1+ r,x_2,x_3,\xi,z),\\
\hat{\phi}_{2,3}:(x_1,x_2,x_3,r,\xi,z) & \longrightarrow  (x_1,x_2,x_2+r,x_3,\xi,z),\\
\hat{\phi}_{3,4}:(x_1,x_2,x_3,r,\xi,z) & \longrightarrow  (x_1,x_2,x_3,x_3+ r,\xi,z).
\end{split}
\end{equation}

Calculating the pullback of the volume form $\sigma$ on $\Cnf[\R;4]\times S^2\times [0,1]$ over $\hat{\phi}_{i,j}$ shows that

\begin{equation}\label{eq:phi-orientations}
\hat{\phi}^\ast_{1,2}\sigma =  -\hat{\phi}^\ast_{2,3}\sigma =  \hat{\phi}^\ast_{3,4}\sigma,
\end{equation}

i.e. $\hat{\phi}^\ast_{1,2}$ and $\hat{\phi}^\ast_{3,4}$ are orientation preserving and $\hat{\phi}^\ast_{2,3}$ is orientation reversing.
Computing the boundary extensions of Gauss maps[extended to the identity on the extra $S^2$ factor.] $(h\times \operatorname{id})\circ \hat{\phi}_{i,j}=(h_{13}\times h_{42}\times \operatorname{id})\circ \hat{\phi}_{i,j}$  along $S_{i,j}=\{r=0\}$ in $\hat{\phi}_{i,j}$ coordinates,  yields

\begin{equation}\label{eq:h-bdry-extension}
\begin{split}
(h\times \operatorname{id}) & \circ\hat{\phi}_{1,2}(x_1,x_2,x_3,0,\xi,z)\\
& =\displaystyle\lim_{r\to0}\left(\frac{K(x_2,z)-K(x_1,z)}{|K(x_2,z)-K(x_1,z)|},\frac{K(x_1+ r,z)-K(x_3,z)}{|K(x_1+ r,z)-K(x_3,z)|},\xi\right),
\end{split}
\end{equation}

\begin{equation}\label{eq:h-bdry-extension-2}
\begin{split}
(h\times \operatorname{id}) & \circ\hat{\phi}_{2,3}(x_1,x_2,x_3,0,\xi,z)\\
& = \left(\frac{K(x_2,z)-K(x_1,z)}{|K(x_2,z)-K(x_1,z)|},\frac{K(x_2,z)-K(x_3,z)}{|K(x_2,z)-K(x_3,z)|},\xi\right),
\end{split}
\end{equation}

\begin{equation}\label{eq:h-bdry-extension-3}
\begin{split}
(h\times \operatorname{id}) & \circ\hat{\phi}_{3,4}(x_1,x_2,x_3,0,\xi,z)\\
& = \left(\frac{K(x_3,z)-K(x_1,z)}{|K(x_3,z)-K(x_1,z)|},\frac{K(x_2,z)-K(x_3,z)}{|K(x_2,z)-K(x_3,z)|},\xi\right).
\end{split}
\end{equation}

Adding the $S^2$ factor, does not change the evaluation of $((\partial \pi)_\ast \beta)_{\mathcal{K}}$. Indeed, using notation in (<ref>), yields for each $S_{i,j}$:

\begin{equation}\label{eq:pi-beta-id-eval}
\begin{split}
((\partial \pi\bigl|_{S_{i,j}})_\ast \beta)_{\mathcal{K}} & =\bigl((\partial \pi\bigl|_{S_{i,j}})_\ast (h_{13}\times h_{42})^\ast (\eta_N\times\eta_N)\bigr)_{\mathcal{K}}\\
& =\bigl((\partial \pi\bigl|_{S_{i,j}\times S^2})_\ast (h_{13}\times h_{42}\times \operatorname{id})^\ast (\eta_N\times\eta_N\times\eta_N)\bigr)_{\mathcal{K}}\\
& =((\partial \pi\bigl|_{S_{i,j}\times S^2})_\ast (\beta\wedge \eta_N))_{\mathcal{K}},
\end{split}
\end{equation}

where the second idenitity follows from the product integration and $\varint_{S^2} \eta_N=1$, also the projection $\pi$ to $[0,1]$ in the last term is properly adjusted by adding the $S^2$ factor in its domain.

§.§ Faces of $\partial \Cnf[\K_N,\R^3;3,1]$ appearing in $((\partial \pi')_\ast \beta')_{\mathcal{K}}$

By (<ref>), $(\partial \pi')_\ast \beta'$is the sum of push forwards of $\beta'$ along the boundary faces of the fiber of

\[
\pi':\Cnf[\K_N,\R^3;3,1]\longrightarrow \K_N,
\]

This boundary has faces indexed by the following

\begin{equation}\label{eq:Cnf[K,R^3;3,1]-faces}
\begin{split}
\textit{ Principal Faces: }&\{1,2\},\{2,3\},\{1,\mathbf{4}\},\{2,\mathbf{4}\},\{3,\mathbf{4}\};\\
\textit{ Hidden Faces: }&\{1,2,3\},\{1,2,\mathbf{4}\},\{2,3,\mathbf{4}\};\\
\textit{ Anomalous Face: }&\{1,2,3,\mathbf{4}\};
\end{split}
\end{equation}

\begin{equation}\label{eq:Cnf[K,R^3;3,1]-faces-2}
\begin{split}
\textit{ Faces at Infinity: }&\{1\to-\infty\},\{1,2\to-\infty\}\{1,2,3\to-\infty\},\{3\to\infty\},\{2,3\to\infty\},\\
&\{1,2,3\to\infty\},\{1\to-\infty \ \& \ \mathbf{4}\to\infty\},\\
&\{1,2\to-\infty\ \& \ \mathbf{4}\to\infty\},\{1,2,3\to-\infty\ \& \ \mathbf{4}\to\infty\},\\
&\{3\to\infty\ \& \ \mathbf{4}\to\infty\},\{2,3\to\infty\ \& \ \mathbf{4}\to\infty\},\\
&\{1,2,3\to\infty\ \& \ \mathbf{4}\to\infty\},\{\mathbf{4}\to\infty\},
\end{split}
\end{equation}

where the indices are associated to the colliding coordinates. The interior of the fiber of $\Cnf[\K_N,\R^3;3,1]$, over $K_0\in \K_N$ is given as

\[
\begin{split}
\Cnf(K_0,\R^3;3,1) & \cong\bigl\{(x_1,x_2,x_3,\mathbf{x}_4)\ |\ (x_1,x_2,x_3)\in \Cnf(\R;3),\\
& \qquad\qquad\qquad\qquad\quad  \mathbf{x}_4\in \R^3-\{K_0(x_1), K_0(x_2), K_0(x_3)\} \bigr\}.
\end{split}
\]

Note that $``\mathbf{4}\to\infty"$ means that the norm of $\mathbf{x}_4\in\R^3$ is tending to $\infty$.

§.§ Computations for $((\partial \pi')_\ast \beta')_{\mathcal{K}}$ along the principal faces
Similarly as in Section <ref>, we first
calculate in local coordinates in the neighborhood of principal faces: $A=\{1,\mathbf{4}\}$, $\{2,\mathbf{4}\}$, $\{3,\mathbf{4}\}$.
The local coordinates about
$S_{i,\mathbf{4}}=\{r=0\}$, amended by the factor $[0,1]$, i.e. the domain of the plot $\mathcal{K}$ in (<ref>) are given as

\[
\begin{split}
\psi_{i,\mathbf{4}}:\Cnf(\R;3)\times [0,\infty)\times S^2\times [0,1] &\longrightarrow \Cnf[\R^3;4]\times [0,1]\\
\psi_{i,\mathbf{4}}: (x_1,x_2,x_3,r,\xi, z) & \longrightarrow (K(x_1,z),K(x_2,z),K(x_3,z), K(x_i,z)+r\xi, z).
\end{split}
\]

A quick computation

\[
dK(x_1,z)\wedge \ldots \wedge (d K(x_i,z)+r\xi)\wedge dz\equiv d x_1\wedge d x_2\wedge d x_3\wedge d(r\xi)\wedge dz
\]

shows that $\psi_{i,\mathbf{4}}$'s do not change the orientation.
Computing the smooth boundary extensions of Gauss maps  along $S_{i,\mathbf{4}}$

\[
h'\circ \psi_{i,\mathbf{4}}=h_{3;1}\circ \psi_{i,\mathbf{4}},
\]

\begin{equation}\label{eq:h'-bdry-extension}
\begin{split}
(h'\circ\psi_{1,\mathbf{4}}) & (x_1,x_2,x_3,0,\xi,z)=\displaystyle\lim_{r\to0}\bigg(\frac{K(x_1,z)+r\xi-K(x_1,z)}{|K(x_1,z)+r\xi-K(x_1,z)|},
\end{split}
\end{equation}

\begin{equation}\label{eq:h'-bdry-extension-1}
\begin{split}
\end{split}
\end{equation}

\begin{equation}\label{eq:h'-bdry-extension-2}
\begin{split}
(h'\circ\psi_{2,\mathbf{4}}) & (x_1,x_2,x_3,0,\xi,z) = \left(\frac{K(x_2,z)-K(x_1,z)}{|K(x_2,z)-K(x_1,z)|},-\xi,\frac{K(x_2,z)-K(x_3,z)}{|K(x_2,z)-K(x_3,z)|}\right),
\end{split}
\end{equation}

\begin{equation}\label{eq:h'-bdry-extension-3}
\begin{split}
(h'\circ\psi_{3,\mathbf{4}}) & (x_1,x_2,x_3,0,\xi,z) =\left(\frac{K(x_3,z)-K(x_1,z)}{|K(x_3,z)-K(x_1,z)|},\frac{K(x_2,z)-K(x_3,z)}{|K(x_2,z)-K(x_3,z)|},\xi\right).
\end{split}
\end{equation}

Comparing, the smooth extensions of the Gauss maps in (<ref>), (<ref>), (<ref>) and (<ref>), (<ref>), (<ref>) yields, up to permutation of the $S^2$ factors and
antipodal symmetry $A$ in (<ref>)

\[
(h\times \operatorname{id})\circ\hat{\phi}_{i,i+1}=h'\circ\psi_{i,\mathbf{4}},\qquad i=1,2,3.
\]

§.§ Canceling of the pushforwards along the principal faces
By Proposition <ref>, and the above computations we obtain the following identities

\begin{equation}\label{eq:principal-faces-cancel}
((\partial \pi\bigl|_{S_{i,i+1}})_\ast \beta)_{\mathcal{K}}=((\partial \pi\bigl|_{S_{i,i+1}\times S^2})_\ast (\beta\wedge \eta_N))_{\mathcal{K}}=((\partial \pi'\bigl|_{S_{i,\mathbf{4}}})_\ast \beta')_{\mathcal{K}},\qquad i=1,2,3.
\end{equation}

Note that for $i=2$ the sign in (<ref>), which appears in (<ref>) is compensated with the orientation of the face $S_{2,3}$ in coordinates $\hat{\phi}_{2,3}$, see (<ref>).

§.§ Vanishing over Hidden and Anomalous Faces of $\Cnf[\R;4]$

Suppose that $S_A$ is a codimension one face of $\Cnf[\R;4]$ indexed by a subset $A\subseteq\{1,2,3,4\}$, $|A|>2$.  We claim that if $A$ contains $\{1,3\}$ or $\{2,4\}$, then

\begin{equation}\label{eq:phi-A-beta}
\phi^\ast_A\beta\bigl|_{r=0}=0,
\end{equation}

this applies to the hidden faces, and the anomalous face listed in (<ref>).
The local parametrization $\phi_{A}$ in (<ref>), where the points are colliding at $K(x_1)$,  composed with the map $h_{13}$ extends along $S_A$ as follows

\[
h_{13}\circ\phi_{A}(x_1,\ldots,u_3,\ldots,r,z)=\lim_{r\to 0} \frac{K(x_1+r u_3,z)-K(x_1,z)}{|K(x_1+r u_3,z)-K(x_1,z)|} =\frac{K'(x_1,z)}{|K'(x_1,z)|},
\]

By  $(ii)$, $\frac{K'(x,z)}{|K'(x,z)|}$ is excluded from the support of $\eta_N$, thus $\phi^\ast_{A}(h_{13}^\ast \eta_N)\bigl|_{r=0}=0$, and therefore $\beta|_{S_{A}}=0$.
Similarly, if $A$ contains $\{2,4\}$  then, the local parametrization $\phi_{A}$ composed with the map $h_{42}$ has the same property, thus $\phi^\ast_{A}(h^\ast_{42}\eta_N))=0$.

§.§ Vanishing over Faces at Infinity  of $\Cnf[\R;4]$

Let us perform a computation for $S_A$ when e.g. $4\to +\infty$ in $A$, and $2\in A$, i.e. $K(x_2)$ is fixed on the knot. For such faces, the local coordinates in (<ref>) differ only by replacing $ru_i$ with $\tfrac{1}{r} u_i$ in the definition. The map $h_{42}$ extends along $S_A$, as follows

\[
\begin{split}
h_{42}\circ\phi_{A}(\ldots,x_2,\ldots,x_4,\ldots,u_4,r,z) & =\lim_{r\to 0} \frac{K(x_2,z)-K(x_4+\tfrac{1}{r} u_4,z)}{|K(x_2,z)-K(x_4+\tfrac{1}{r} u_4,z)|},\\
& =\lim_{r\to 0} \frac{K(x_2,z)- \frac{1}{r} X}{|K(x_2,z)- \frac{1}{r} X|}=X
\end{split}
\]

where $X$ is the direction vector orthogonal to $N$, defined in the condition  $(iii)$. By assumption $(ii)$, $X$ is excluded from the support of $\eta_N$, thus $h_{42}^\ast(\eta_N)=0$, and therefore[For small enough $r$ the image of $h_{42}$ is outside of the support of $\eta_N$.] $\phi^\ast_{A}\beta|_{S_{A}}=\phi^\ast_{A}(h_{41}^\ast\eta_N\wedge h_{24}^\ast\eta_N\wedge h_{43}^\ast\eta_N)\bigr|_{r=0}=0$. Similarly, if $A$ contains $\{2\to +\infty,4\to +\infty\}$  then in the above limit, we replace $K(x_2,z)$ with $K(x_2+\tfrac{1}{r} u_2,z)$ and using $(iii)$ yields the same result.  The remaining cases
are fully analogous, just by changing indexes of coordinates.

§.§ Vanishing over faces $\{1,2\}$, $\{2,3\}$ of $\Cnf[\K_N,\R^3;3,1]$

The local coordinates around $S_{\{1,2\}}=\{r=0\}$ is given as

\[
\begin{split}
\psi_{1,2}:\Cnf(\R;2)\times [0,\infty)\times (\R^3-\{K(x_1,z)\})\times [0,1] &\longrightarrow \Cnf[\R^3;4]\times [0,1]\\
\psi_{1,2}: (x_1,x_2,r,\bx, z) \longrightarrow (K(x_1,z), K(x_1+r,z), & K(x_2,z),  \bx, z).
\end{split}
\]

\begin{equation}\label{eq:h'-bdry-extension-5}
\begin{split}
(h'\circ\psi_{1,2}) & (x_1,x_2,0,\bx, z)=\displaystyle\lim_{r\to0}\bigg(\frac{\bx-K(x_1,z)}{|\bx-K(x_1,z)|},\frac{K(x_1+r,z)-\bx}{|K(x_1+r,z)-\bx|},\frac{\bx-K(x_2,z)}{|\bx-K(x_2,z)|}\bigg)\\
\end{split}
\end{equation}

Clearly, the first two factors are equal after applying the antipodal map $A$, (<ref>),  ($h_{24}=A\circ h_{42}$). Therefore,

\[
\begin{split}
\psi_{1,2}^\ast (h^\ast_{1\mathbf{4}}\eta_N)\bigl|_{r=0} =\psi_{1,2}^\ast (h^\ast_{\mathbf{4}2} A^\ast\eta_N)|_{r=0} & =-\psi_{1,2}^\ast (h^\ast_{\mathbf{4}2} \eta_S)|_{r=0},\qquad \text{and}\qquad\\
\psi_{1,2}^\ast\beta'|_{r=0}=\psi_{1,2}^\ast (h^\ast_{1\mathbf{4}}\eta_N\wedge h^\ast_{\mathbf{4}2} \eta_N\wedge h^\ast_{\mathbf{4}3} \eta_N)|_{r=0} & =-\psi_{1,2}^\ast (h^\ast_{\mathbf{4}2} (\eta_S\wedge \eta_N)\wedge h^\ast_{\mathbf{4}3} \eta_N)|_{r=0}=0,\\
\end{split}
\]

where the vanishing in the last identity follows since $\eta_S\wedge \eta_N=0$ on $S^2$ as the degree $4$ form. For the face $S_{\{2,3\}}$ and $S_{\{1,2,3\}}$ the computation is analogous.

§.§ Vanishing over the hidden and anomalous faces  of $\Cnf[\K_N,\R^3;3,1]$

All of the faces, under consideration, contain the indices $\{i,j,\mathbf{4}\}$. Without loss of generality, we present calculation for $A=\{1,2,\mathbf{4}\}$, the local coordinates around $S_{\{1,2,\mathbf{4}\}}=\{r=0\}$ are given as

\[
\begin{split}
\psi_A:\Cnf(\R;2)\times S^3\times [0,\infty)\times [0,1] &\longrightarrow \Cnf[\R^3;4]\times [0,1]\\
\psi_A: (x_1,x_2,u,\xi, r, z) \longrightarrow (K(x_1,z), K(x_1+r u,z), & K(x_2,z), K(x_1,z) +r\xi, z),
\end{split}
\]

(where $(u,\xi)\in S^3$, i.e. $u^2+|\xi|^2=1$, see Appendix <ref> for the general case),

\begin{equation}\label{eq:h'-bdry-extension-4}
\begin{split}
(h'\circ\psi_A) & (x_1,x_2,u,\xi, 0, z)\\
& = \displaystyle\lim_{r\to 0}\bigg(\frac{(K(x_1,z) + r\xi)-K(x_1,z)}{|(K(x_1,z) +r\xi)-K(x_1,z)|},\\
&\qquad \frac{K(x_1+ r u,z)-(K(x_1,z) +r\xi)}{|K(x_1+r u,z)-(K(x_1,z) +r\xi)|}, \frac{(K(x_1,z) +r\xi)-K(x_2,z)}{|(K(x_1,z) +r\xi)-K(x_2,z)|}\bigg)\\
&=\left(\frac{\xi}{|\xi|},\frac{K'(x_1,z) u-\xi}{|K'(x_1,z) u-\xi|},\frac{K(x_1,z)-K(x_2,z)}{|K(x_1,z)-K(x_2,z)|}\right).
\end{split}
\end{equation}

Let us estimate how much the image of the above map intersects the support of $\eta_N\times \eta_N\times \eta_N$, which is a product of small $\varepsilon$–disks around $N\in S^2$ of each $S^2$ factor in the codomain of $h'$ in (<ref>). Thanks to <ref>,
we obtain[where $\frac{1}{\varepsilon} O(\varepsilon)\longrightarrow \text{const}$ for $\varepsilon\longrightarrow 0$.]

\[
\frac{\xi}{|\xi|}=N+O(\varepsilon),\qquad \frac{K'(x_1,z) u-\xi}{|K'(x_1,z) u-\xi|}=N+O(\varepsilon),
\]

which yields

\[
\begin{split}
\xi & =|\xi| N+O(\varepsilon),\qquad K'(x_1,z) u-\xi=|K'(x_1,z) u-\xi| N+O(\varepsilon),\\
K'(x_1,z) u & = (|\xi|+|K'(x_1,z) u-\xi|) N+O(\varepsilon).
\end{split}
\]

Taking the cross product $\times N$ both sides of the last identity

\[
(K'(x_1,z)\times N)  u  = O(\varepsilon),\quad \Rightarrow\quad u  = O(\varepsilon),
\]

since  $0<|K'(x_1,z)\times N|<\infty$, and $|u|\leq 1$.  In conclusion, we obtain for small $\varepsilon$

\[
(h'\circ\psi_A)(x_1,x_2,u,\xi, 0, z)\approx \left(\frac{\xi}{|\xi|},-\frac{\xi}{|\xi|},\frac{K(x_1,z)-K(x_2,z)}{|K(x_1,z)-K(x_2,z)|}\right),
\]

thus the first two coordinates, certainly have no intersection with the support of the first wo factors of $\eta_N\times \eta_N\times \eta_N$ and therefore $ \psi^\ast_A\beta'\bigl|_{r=0}=0$ as claimed.

§.§ Vanishing over the faces at infinity of $\Cnf[\K_N,\R^3;3,1]$

Arguments for vanishing are analogous to those presented in Section <ref>.
Let us perform the computation for $S_A$ when e.g. ${\bf 4}\to \infty$ in $A$, and $i\in A$, $1\leq i\leq 3$, i.e. $K(x_i)$ moves to $\pm\infty$ along the knot, and $\bx_{4}\to \infty$ in $\R^3$. The map $h_{{\bf 4} i}$ extends along $S_A$, as follows

\[
\begin{split}
h_{{\bf 4} i}\circ\phi_{A}(\ldots,x_i,\ldots,u_i\ldots, 0,\xi,z) & =\lim_{r\to 0} \frac{\frac{1}{r}\xi-K(x_i+\frac{1}{r} u_i,z)}{|\frac{1}{r}\xi-K(x_i+\frac{1}{r} u_i,z)|}\\
=\lim_{r\to 0} \frac{\frac{1}{r}\xi-\frac{1}{r}  X}{|\frac{1}{r}\xi-\frac{1}{r}  X|}=\frac{\xi- X}{|\xi- X|}.
\end{split}
\]

where$X$ is the direction vector orthogonal to $N$, defined in the condition  $(iii)$. Choosing $c$ large enough, i.e. reparametrizing the knot far away from zero along $\R$, we may assure that for $r\ll 1$: $\operatorname{im}({h_{{\bf 4} i}\circ\phi_{A}})\cap \supp{\eta_N}=\varnothing$, which guarranties that for $r\ll 1$,
$\phi^\ast_{A}(h_{{\bf 4} i}^\ast \eta_N)\bigl|_{r=0}=0$. In the case only ${\bf 4}\to \infty$ in $A$, i.e. all $K(x_i)$, $1\leq i\leq 3$ are fixed on the knot, we have

\[
\begin{split}
(h_{{\bf 4} 1}\times h_{2 {\bf 4}}\times h_{{\bf 4} 3}) & \circ\phi_{A}(x_1,x_2,x_3, 0,\xi,z)   = \displaystyle\lim_{r\to 0}\bigg(\frac{ \frac{1}{r}\xi-K(x_1,z)}{|\frac{1}{r}\xi-K(x_1,z)|},\frac{K(x_2,z)-\frac{1}{r}\xi}{|K(x_2,z)-\frac{1}{r}\xi|},\\
&\qquad  \frac{ \frac{1}{r}\xi-K(x_3,z)}{|\frac{1}{r}\xi-K(x_3,z)|}\bigg)=\big(\xi,-\xi,\xi\big),
\end{split}
\]

again the pullback $\phi^\ast_{A}((h_{{\bf 4} 1}\times h_{2 {\bf 4}}\times h_{{\bf 4} 3})^\ast(\eta_N\times \eta_N\times \eta_N)\bigl|_{r=0}=0$, because the supports of $\phi^\ast_{A} h_{{\bf 4} 1}^\ast \eta_N$ (or $\phi^\ast_{A} h_{{\bf 4} 3}^\ast \eta_N$) and $\phi^\ast_{A} h_{2 {\bf 4}}^\ast \eta_N$  are disjoint for $r\ll 1$.

The only case left is $\{ {\bf 4}\to \infty\} \not\in A$, and $\{i\to \pm \infty\}\in A$, when

\[
\begin{split}
h_{{\bf 4} i}\circ\phi_{A}(\ldots,x_i,\ldots,u_i\ldots, 0,\bx,z) & =\lim_{r\to 0} \frac{\bx-K(x_i\pm\frac{1}{r} u_i,z)}{|\bx-K(x_i\pm \frac{1}{r} u_i,z)|}=\lim_{r\to 0} \frac{\bx-\frac{1}{r} c X}{|\bx-\frac{1}{r} c X|}=\pm\frac{ X}{|X|}.
\end{split}
\]

thus for $r\ll 1$,  $\operatorname{im}({h_{{\bf 4} i}\circ\phi_{A}})\cap \supp{\eta_N}=\varnothing$, which guaranties that
$\phi^\ast_{A}(h_{{\bf 4} i}^\ast \eta_N)\bigl|_{r=0}=0$.

§.§ Finishing the proof of invariance of $I(K)$

Combining the results of Sections <ref> – <ref>

\[
d I(K(\,\cdot\, ,z)) =d \bigl((\pi_\ast\beta)_{\mathcal{K}} -(\pi'_\ast\beta')_{\mathcal{K}}\bigr)
=  -((\partial \pi)_\ast \beta)_{\mathcal{K}} +((\partial \pi')_\ast \beta')_{\mathcal{K}}
\]

proves invariance of the integral in (<ref>) of Theorem <ref>.

§.§ Arrow diagram counting for double crossing projections

Recall the standard argument <cit.>, tells us that for any map $F:M^m\longrightarrow N^n$, $m=n$, between manifolds (with corners) and a regular value $y\in \operatorname{int}(N)$, where $F^{-1}(y)=\{x_1,\ldots,x_m\}\subset \operatorname{int}(M)$, the local degree [20] of $F$ at $y$ is given as

\[
\deg_y F = \sum_{i} \operatorname{sign} \bigl(\operatorname{det} DF(x_i)\bigr)=\int_M F^\ast\eta_y,
\]

where $\eta_y$ is a unit $n$–form ($\varint_N \eta_y=1$), supported on a sufficiently small neighborhood of $y$ in $N$.
Clearly, this argument was used in Section <ref> to obtain the standard formula for the linking number in (<ref>). For the long knot, which is near its regular planar projection we have

Suppose $K$ projects on the plane orthogonal to $N$ in a regular double crossing knot diagram, then the integral $I(K)$ in (<ref>) equals

\begin{equation}\label{eq:I(K)-gauss-dbl-crossing}
I(K)=\varint\limits_{\Cnf[\R;4]} h^\ast_{13}\eta_N\wedge h^\ast_{42}\eta_N=\langle \vvcenteredinclude{.3}{gd_c2_chord.pdf},G_K\rangle.
\end{equation}

Since $\langle \vvcenteredinclude{.3}{gd_c2_chord.pdf},G_K\rangle$ coincides with the arrow diagram formula of [29] for the
second coefficient of the Conway polynomial $c_2(K)$, and $I(K)$ is the knot invariant, we have

Clearly, we may isotope $K$ so that it coincides with the image of its projection except at the double crossings, where we can create arbitrary “small” overpasses.

For the Gauss map

\[
h =h_{13}\times h_{42}:\Cnf(\R;4)\longrightarrow S^2\times S^2,
\]

the inverse image of $(N,N)$ under $h$ is given as

\[
h^{-1}(N,N)=\{(x_1,x_2,x_3,x_4)\ |\ \tfrac{K(x_3)-K(x_1)}{|K(x_3)-K(x_1)|}=N, \tfrac{K(x_2)-K(x_4)}{|K(x_2)-K(x_4)|}=N\}.
\]

Since the crossings are transverse double crossings, $(N,N)$ is the regular value and the local degree formula, reviewed at the beginning of this section yields

\[
\varint\limits_{\Cnf[\R;4]} h^\ast_{13}\eta_N\wedge h^\ast_{42}\eta_N=\langle \vvcenteredinclude{.3}{gd_c2_chord.pdf},G_K\rangle.
\]

To finish the proof of (<ref>), we need to show

\begin{equation}\label{eq:tripod-vanish}
\varint\limits_{\Cnf[\K_N,\R^3;3,1]} h^\ast_{1{\bf 4}}\eta_N\wedge h^\ast_{{\bf 4}2}\eta_N\wedge h^\ast_{3{\bf 4}}\eta_N=0.
\end{equation}

Using the formula for $\eta_N$ in (<ref>), a direct computation yields

\[
\begin{split}
& \beta'=h^\ast_{1{\bf 4}}\eta_N\wedge h^\ast_{{\bf 4}2}\eta_N\wedge h^\ast_{3{\bf 4}}\eta_N=\left[\prod_{i=1}^3f_\varepsilon\left((-1)^{i-1}\frac{\mathbf{x}_4-K(x_i)}{|\mathbf{x}_4-K(x_i)|}\right)\right]\\
&\left[\sum_{\sigma\in S_3}\left(\textrm{sgn}(\sigma)\bigwedge_{i=1}^3\frac{\langle (\mathbf{x}_4-K(x_i))\times K'(x_i),e_{\sigma(i)}\rangle}{|\mathbf{x}_4-K(x_i)|^3} dx_i\wedge dv_{\sigma(i)}\right)\right]\\
&=\left[\prod_{i=1}^3f_\varepsilon\left((-1)^{i-1}\frac{\mathbf{x}_4-K(x_i)}{|\mathbf{x}_4-K(x_i)|}\right)\right]\bigg[\textrm{det}[E_1,-E_2,E_3] d x_1\wedge d x_2\wedge d x_3\wedge d \mathbf{x}_4\bigg],\\
\end{split}
\]

where $\mathbf{x}_4=(v_1,v_2,v_3)$, $d \mathbf{x}_4=dv_1\wedge dv_2\wedge dv_3$, $e_i$ denotes the standard $i$th basis vector in $\R^3$ and

\begin{equation}\label{eq:E_i}
E_i=E_i(\mathbf{x}_4,x_i)=\frac{(\mathbf{x}_4-K(x_i))\times K'(x_i)}{|\mathbf{x}_4-K(x_i)|^3},\ i=1,2,3.
\end{equation}

Since, the support of $\eta_N$ is a small $\varepsilon$–ball around $N$, $x_i$, $\mathbf{x}_4$, $i=1,2,3$ need to satisfy[$\approx_\varepsilon$ stands for $\varepsilon$–close in the standard round metric on $S^2$]

\[
\tfrac{\mathbf{x}_4-K(x_1)}{|\mathbf{x}_4-K(x_1)|}\approx_\varepsilon N,\qquad \tfrac{K(x_2)-\mathbf{x}_4}{|K(x_2)-\mathbf{x}_4|}\approx_\varepsilon N,\qquad \tfrac{\mathbf{x}_4-K(x_3)}{|\mathbf{x}_4-K(x_3)|}\approx_\varepsilon N,
\]

and therefore, the only possible configuration for $K(x_i)$, $\mathbf{x}_4$, $i=1$,$2$,$3$ occurs in the neighborhood of the double crossing as illustrated on Figure <ref>, where the arrows are all pointing into directions $\varepsilon$–close to $N$.

Configuration of points in $\Cnf[\K_N,\R^3;3,1]$, whose image under $h'$ is close to

W.l.o.g. we assume that the strands are straight line segments in the neighborhood of the crossings (as pictured on Figure <ref>). Since $K(x_3)$ and $K(x_1)$ are on the same line  (lower strand) $\ell_{\operatorname{lower}}$, in the direction of $K'(x_1)$ and $K'(x_3)$,
vectors $\mathbf{x}_4-K(x_1)$, $\mathbf{x}_4-K(x_3)$ are either parallel or span the plane $P$ containing the line  $\ell_{\operatorname{lower}}$. In turn,
$(\mathbf{x}_4-K(x_1))\times K'(x_1)$ and $(\mathbf{x}_4-K(x_3))\times K'(x_3)$ must be both orthogonal to $P$, and as a consequence vectors $E_1$ and $E_3$ in (<ref>) are parallel and $\det[E_1,-E_2,E_3]=0$, proving (<ref>).

§ PROOF OF THEOREM <REF>

The remaining part is an extension of the formula (<ref>) to knot diagrams with multiple crossings given in (<ref>):

\begin{equation}\label{eq:gd_c2_mult-crossings-2}
\begin{split}
c_2(K) & =\langle \vvcenteredinclude{.3}{gd_c2_chord.pdf},G_K\rangle+ \tfrac{1}{2}\bigl(\langle \vvcenteredinclude{.3}{gd_c2_chord-left.pdf},G_K\rangle\\
&\qquad +\langle \vvcenteredinclude{.3}{gd_c2_chord-mid.pdf},G_K\rangle+\langle \vvcenteredinclude{.3}{gd_c2_chord-right.pdf},G_K\rangle\bigr).
\end{split}
\end{equation}

Rather than obtaining this extension from evaluating the integral in (<ref>), we provide a combinatorial argument.

A triple crossing perturbation and corresponding fragments of the Gauss diagram.

First, for simplicity we think of a long knot $K$ and its planar diagram as the same object (i.e. the parametrization of $K$ is almost entirely in the plane of the projection except small bridges over crossings). If $K$ is a (transverse) multi-crossing knot, we denote by $K'$ its local perturbation. Specifically, $K'$ has a regular double crossing diagram which agrees with $K$ outside small balls containing the multiple crossings of $K$, and is isotopic to $K$, Figure <ref>.
For that reason, we trivially observe that

\begin{equation}\label{eq:c_2-via-K'}
c_2(K)=\langle \vvcenteredinclude{.3}{gd_c2_chord.pdf},G_{K'}\rangle=\sum_{\phi':\vvcenteredinclude{.15}{gd_c2_chord.pdf}\longrightarrow G_{K'}} \operatorname{sign}(\phi').
\end{equation}

Clearly, the goal is to replace the sum over embeddings

\begin{equation}\label{eq:embeddings_in_G_K'}
\mathcal{E}(G_{K'}) =\{\phi':\vvcenteredinclude{.2}{gd_c2_chord.pdf}\longrightarrow G_{K'}\}
\end{equation}

with a sum over embeddings
into $G_K$. Denote by $e(G_K)$ and $e(G_{K'})$ the set of oriented and signed chords of $G_K$ and $G'_K$ respectively, each embedding $\phi'\in \mathcal{E}(G_{K'})$ can be identified with a choice of the pair of distinct chords from $e(G_{K'})$, i.e. $\phi'=\{\alpha',\beta'\}$, $\alpha',\beta'\in e(G_{K'})$, satisfying the obvious condition on their endpoints. Therefore, the sum in (<ref>) can be written as a sum over unordered pairs of chords

\[
\mathcal{E}^2(G_{K'})  =\{\{\alpha',\beta'\}\ |\ \alpha',\beta'\in e(G_{K'}), \alpha'\neq\beta'\},\\
\]

as follows

\begin{equation}\label{eq:c_2-via-K'-chords}
\langle \vvcenteredinclude{.3}{gd_c2_chord.pdf},G_{K'}\rangle=\sum_{\{\alpha',\beta'\}\in \mathcal{E}^2(G_{K'})} f_{\vvcenteredinclude{.1}{gd_c2_chord.pdf}}(\alpha',\beta'),
\end{equation}

where $f_{\vvcenteredinclude{.1}{gd_c2_chord.pdf}}$ is an obvious function (valued in $\{0,\pm 1\}$).
Next, observe that, by construction of $K'$, we have a bijection

\[
F: e(G_{K'})\longrightarrow e(G_K),
\]

which preserves the orientation of chords and their signs. Further, for each $\phi'=\{\alpha',\beta'\}\in \mathcal{E}(G_{K'})$, we will use notation

\[
\phi=\{\alpha,\beta\}=F\circ \phi'=\{F(\alpha'),F(\beta')\},
\]

under the above identification of pairs of chords and embeddings.

There are two cases of interest for such $\phi$: first $\alpha$ and $\beta$ do not share any endpoints then $\phi$ embeds $\vvcenteredinclude{.2}{gd_c2_chord.pdf}$ into $G_K$ (i.e. $\phi$ is a regular embedding), second $\alpha$ and $\beta$ share an endpoint, then $\phi$ embeds one of

\begin{equation}\label{eq:diag-common-ends}
\vvcenteredinclude{.3}{gd_c2_chord-left.pdf},\quad \vvcenteredinclude{.3}{gd_c2_chord-mid.pdf},\quad
\vvcenteredinclude{.3}{gd_c2_chord-right.pdf}
\end{equation}

into $G_K$. In the view, of the above correspondence between the embeddings and the pairs of chords, let us define the following subsets of pairs of chords partitioning $e(G_{K'})^2$:

\begin{equation}\label{eq:E^2-bullet}
\begin{split}
\mathcal{E}^2_\bullet(G_{K'}) & =\{\{\alpha',\beta'\}\in \mathcal{E}^2(G_{K'})\ |\ \alpha\ \text{and}\ \beta\ \text{share an endpoint}\},\\
\mathcal{E}^2_\circ(G_{K'}) & = \{\{\alpha',\beta'\}\in \mathcal{E}^2(G_{K'})\ |\ \alpha\ \text{and}\ \beta\ \text{do not share an endpoint}\},
\end{split}
\end{equation}

the sum in (<ref>) splits accordingly as

\begin{equation}\label{eq:c_2-via-K'-chords-2}
\langle \vvcenteredinclude{.3}{gd_c2_chord.pdf},G_{K'}\rangle=\sum_{\{\alpha',\beta'\}\in \mathcal{E}^2_\circ(G_{K'})} f_{\vvcenteredinclude{.1}{gd_c2_chord.pdf}}(\alpha',\beta')+\sum_{\{\alpha',\beta'\}\in \mathcal{E}^2_\bullet(G_{K'})} f_{\vvcenteredinclude{.1}{gd_c2_chord.pdf}}(\alpha',\beta').
\end{equation}

Note that the first term in this sum equals (by definition)

\begin{equation}\label{eq:sum-E^2-circ}
\sum_{\{\alpha',\beta'\}\in \mathcal{E}^2_\circ(G_{K'})} f_{\vvcenteredinclude{.1}{gd_c2_chord.pdf}}(\alpha',\beta')=\langle \vvcenteredinclude{.3}{gd_c2_chord.pdf},G_{K}\rangle.
\end{equation}

The expression $\langle \vvcenteredinclude{.25}{gd_c2_chord.pdf}+  \vvcenteredinclude{.25}{gd_c2_chord-left.pdf}+ \vvcenteredinclude{.25}{gd_c2_chord-mid.pdf}+\vvcenteredinclude{.25}{gd_c2_chord-right.pdf},G_K\rangle$ does not yield a correct count for $c_2(K)$,
because there is no one–to–one correspondence between the embeddings of  diagrams with common endpoints listed in (<ref>) in $G_K$, with a subset of $\mathcal{E}(G_{K'})$ in (<ref>), preserving signs and orientations, see Example (<ref>).

Verifying (<ref>) on triple–crossings.

Triple Crossing     Arrow diagrams contributing to (<ref>)     Perturbed Crossing     Count in (<ref>)
0.3     0.3 $+$ 0.3 $\rightsquigarrow 2$     0.3       1

0.3     0.3 $+$ 0.3  $\rightsquigarrow 0$     0.3      0

0.3     0.3 $+$ 0.3  $\rightsquigarrow -2$     0.3      -1

0.3     0.3 $+$ 0.3  $\rightsquigarrow 0$     0.3      0

0.3     0.3 $+$ 0.3  $\rightsquigarrow 0$     0.3      0

0.3     0.3 $+$ 0.3  $\rightsquigarrow -2$     0.3     -1

0.3     0.3 $+$ 0.3  $\rightsquigarrow 0$     0.3     0

0.3     0.3 $+$ 0.3  $\rightsquigarrow 2$     0.3      1

In order to find an expression for the second term in (<ref>) note that for every triple of distinct chords $\alpha',\beta',\gamma'\in e(G_{K'})$, either $\{\alpha,\beta,\gamma\}=F\{\alpha',\beta',\gamma'\}$ form a triple crossing[where the labels $\alpha$, $\beta$, $\gamma$ can be permuted and arrow directions with signs assigned freely.] $T=\vvcenteredinclude{.3}{triangle-gd-abg.pdf}$ in $G_K$ or not. For each triple crossing the set of pairs $\{\alpha,\beta\}$, $\{\alpha,\gamma\}$, $\{\gamma,\beta\}$ uniquely determine the triple–crossings $T$, i.e. none of the pairs can be in a different triple–crossing. Also every pair $\{\alpha,\beta\}$ in $\mathcal{E}^2_\bullet(G_{K'})$ is a part of a unique triple–crossing. As a result the set $\mathcal{E}^2_\bullet(G_{K'})$ partitions into three elements subsets $\{\alpha',\beta'\}$, $\{\alpha',\gamma'\}$, $\{\gamma',\beta'\}$ which are pairs such that the corresponding $\{\alpha,\beta\}=F\{\alpha',\beta'\}$, $\{\alpha,\gamma\}=F\{\alpha',\gamma'\}$, $\{\gamma,\beta\}=F\{\gamma',\beta'\}$ constitute a triple–crossing. Denoting the set of “triple crossings” by $\mathcal{T}(G_{K'})$, we obtain

\begin{equation}\label{eq:partitioned-sum}
\sum_{\{\alpha',\beta'\}\in \mathcal{E}^2_\bullet(G_{K'})} f_{\vvcenteredinclude{.1}{gd_c2_chord.pdf}}(\alpha',\beta')= \sum_{\{\alpha',\beta',\gamma'\}\in \mathcal{T}(G_{K'})} \bigl(f_{\vvcenteredinclude{.1}{gd_c2_chord.pdf}}(\alpha',\beta')+f_{\vvcenteredinclude{.1}{gd_c2_chord.pdf}}(\alpha',\gamma')+f_{\vvcenteredinclude{.1}{gd_c2_chord.pdf}}(\gamma',\beta')\bigr).
\end{equation}

Table <ref> shows that for each $\{\alpha',\beta',\gamma'\}\in \mathcal{T}(G_{K'})$ the corresponding term in the above sum equals:

\begin{equation}\label{eq:triple-cross-count}
\begin{split}
& f_{\vvcenteredinclude{.1}{gd_c2_chord.pdf}}(\alpha',\beta')+f_{\vvcenteredinclude{.1}{gd_c2_chord.pdf}}(\alpha',\gamma')+f_{\vvcenteredinclude{.1}{gd_c2_chord.pdf}}(\gamma',\beta')=\\
& \qquad\tfrac{1}{2}\bigl\langle \vvcenteredinclude{.3}{gd_c2_chord-left.pdf}+ \vvcenteredinclude{.3}{gd_c2_chord-mid.pdf}+\vvcenteredinclude{.3}{gd_c2_chord-right.pdf}, G_K\bigr\rangle.%\cdots \vvcenteredinclude{.3}{triangle-gd-abg.pdf}\cdots \bigr\rangle.
\end{split}
\end{equation}

The table shows all cases with the top arrow pointing right $\vvcenteredinclude{.3}{triangle-tab-right-cases.pdf}$, all cases with the top arrow pointing left $\vvcenteredinclude{.3}{triangle-tab-left-cases.pdf}$ is obtained analogously by replacing diagrams $\vvcenteredinclude{.3}{gd_c2_chord-left-tab.pdf}$ in the second column with $\vvcenteredinclude{.3}{gd_c2_chord-right-tab.pdf}$. Note that
$\vvcenteredinclude{.3}{triangle-gd-right-invalid.pdf}$ and $\vvcenteredinclude{.3}{triangle-gd-left-invalid.pdf}$ are invalid subdiagrams of $G_K$, thus they do not need consideration. A choice of the local crossing
perturbation in the third column of Table <ref> is irrelevant, since all of them are isotopic[they happen to coincide with the third Redeimeister move.].

Further, the entire sum in (<ref>) can be replaced with

\[
\tfrac{1}{2}\bigl\langle \vvcenteredinclude{.3}{gd_c2_chord-left.pdf}+ \vvcenteredinclude{.3}{gd_c2_chord-mid.pdf}+\vvcenteredinclude{.3}{gd_c2_chord-right.pdf}, G_K\bigr\rangle,
\]

because each pair $\{\rho',\xi'\}\in \mathcal{E}^2_\bullet(G_{K'})$ is a part of a unique triple–crossing $\{\rho,\xi,\psi\}$ in $G_K$. From (<ref>), (<ref>),  (<ref>) and  (<ref>) the formula (<ref>) follows.

§ COMPUTING $C_2$ FROM A PEDAL DIAGRAM

Any knot $K$ has an über–crossing projection, a projection which only has one crossing (called the  übercrossing) and for which each strand bisects this crossing. The concept was coined in  [3] by Adams et al show that every knot has a pedal diagram - an übercrossing projection which no loop is nested in the interior of another, see also [1, 2, 4, 15, 17].

Given a pedal diagram $P$ of a oriented knot $K$ with $n$ loops, follow $P$ in the direction of the orientation, starting with the top strand of the  übercrossing.  By recording the order in which the levels of the  übercrossing are traversed, one obtains a permutation of $1,\dots,n$ which begins with 1.  Together, with the orientation, this permutation completely identifies $K$.  Although the permutation is not necessarily unique, we may use it to compute the multiple-crossing Gauss diagram $G_P$ for $P$, and from $G_P$ compute $c_2(K)$.  We describe this process below for an oriented knot whose pedal diagram is oriented counterclockwise; in the case that the pedal diagram is oriented clockwise, small changes to the process of determining the associated long knot and determining the signs of the chords in the multi-crossing yield a similar result.

Let $K$ be an oriented knot $K$, whose pedal diagram $P$ has $n$ loops, is oriented counterclockwise, and has associated permutation $(1, a_2, a_3,\dots, a_n)$.
Consider the projection $P$ to sit within the cube $-1\leq x,y,z\leq1$ in $\R^3$.  $P$ can be transformed into a long knot diagram with one multi-crossing by cutting the petal labeled with $1$ and $a_2$.   Deform[If $P$ is oriented clockwise, reverse this. The associated multiple crossing Gauss diagram will have $n$ common endpoints, labeled, in order from left to right, as $1,a_2,a_{3},\dots,a_n$] $P$ so that the severed endpoint of strand $1$ is sent to $(\infty,0,0)$ and the severed end point of strand $a_2$ is sent to  $(-\infty,0,0)$.  A multiple crossing Gauss diagram for this long knot has $n$ common endpoints, labeled, in order from left to right, as $a_2,a_{3},\dots,a_n,1$.

The following oriented copy of $4_1$ is associated to the permutation $(1, 3, 5, 2, 7, 4, 6)$.

Every pair $a_i,a_j$ (including $a_1=1$) shares an arrow, $\alpha_{a_i,a_j}$, where the arrow points from max$\{a_i,a_j\}$ to min$\{a_i,a_j\}$, ie, pointing from the lower level strand to the higher level strand in the crossing.  The sign for each arrow is given by

\[
sgn(\alpha_{a_i,a_j})=\begin{cases}(-)^{j-i+1}& \textrm{ if }a_i<a_j\\
(-)^{j-i}& \textrm{ if }a_i<a_j\end{cases}
\]

The value $j-i$ indicates how many strands are in between strand $a_i$ and $a_j$ in the pedal projection.  In a pedal diagram oriented counterclockwise, if $j-i$ is odd, then the strands are traversed in opposite directions in the projection, and if this number is even, then they are traversed in the same direction. If $a_i$ is less than $a_j$, then the $i^{th}$ strand is projected on top of the $j^{th}$, and vice versa. This leads to the four possible cases in Figure  <ref>, which leads to the above formula[If $P$ is oriented clockwise, then these conditions are reversed.] for $sgn(\alpha_{a_i,a_j})$.   For example, in Figure <ref>, $a_1=1$ and $a_2=3$, $2-1=1$; we see that strand $1$ is traversed pointing upwards, and strand $3$ is traversed going downwards.  This leads to a negative crossing between strand $1$ and strand $3$.   Likewise, when computing the sign of the crossing involving strand $1$ and strand $5$, $a_3=5$, and $3-1=2$; we see that both strand $1$ and strand $5$ are traversed pointing upwards.  This leads to a positive crossing.

Possible configurations for strands within a multicrossing. Top row has $a_i<a_j$, so that $a_i$ is the over strand and the bottom row has $a_j<a_i$.  The left column has $j-i$ is odd, and the right column has $j-i$ is even.

Once we have the multiple crossing Gauss Diagram, we count the relevant subdiagrams.  Let $1\leq i<j<k\leq n$;  the subdiagram on $a_i,a_j$ and $a_k$ is equal to one of the triple crossing diagrams $\vvcenteredinclude{.3}{triangle-tab-right-cases.pdf}$ or $\vvcenteredinclude{.3}{triangle-tab-left-cases.pdf}$ found in Table  <ref> only if $a_j<a_i$ and $a_j<a_k$.  Therefore, one may iterate through the ordered common endpoints of the Gauss diagram $\{a_2, \dots, a_n, 1\}$ and find all such triples.

For each such triple:

* If $sgn(\alpha_{a_i,a_j})=sgn(\alpha_{a_i,a_k})=sgn(\alpha_{a_j,a_k})$, then add $1$ to the Casson count.

* Else if $a_k<a_i$ and $sgn(\alpha_{a_i,a_j})=sgn(\alpha_{a_i,a_k})$ and $sgn(\alpha_{a_i,a_k})=-sgn(\alpha_{a_j,a_k})$, then add $-1$ to the Casson count.

* Else if $a_k>a_i$ and $sgn(\alpha_{a_j,a_k})=sgn(\alpha_{a_i,a_k})$ and $sgn(\alpha_{a_i,a_k})=-sgn(\alpha_{a_i,a_j})$, then add $-1$ to the Casson count.

* Else add $0$ to the count.

In addition to contributions from triple crossing diagrams, there are also contributions from quadruples of strands which represent embeddings of $\vvcenteredinclude{.3}{gd_c2_chord.pdf}$ in the multi-crossing.  Let $1\leq i<j<k<l\leq n$; the subdiagram on $a_i,a_j, a_k$, and $a_l$ only if $a_k<a_i$ and $a_j<a_l$.  Again, all such quadruples can be found; if $sgn(\alpha_{a_i,a_k})=sgn(\alpha_{a_j,a_l})$, then $1$ is added to the Casson count, and if $sgn(\alpha_{a_i,a_k})=-sgn(\alpha_{a_j,a_l})$, then $-1$ is added to the Casson count.

For the pedal digram given in <ref>, the associated Gauss diagram will have common endpoints labeled in order from left to right as $3,5,2,7,4,6,1$.  The relevant triples crossing diagrams in this Gauss diagram will be given by strands $\{3,2,7\}$, $\{3,2,4\}$, $\{3,2,6\}$, $\{5,2,7\}$, $\{5,2,4\}$, $\{5,2,6\}$, $\{5,4,6\}$, and $\{7,4,6\}$.  The triples $\{3,2,7\}$, $\{3,2,6\}$, and $\{5,2,4\}$ contribute 1 each, the triple  $\{3,2,4\}$ contributes $-1$, and the other $4$ contribute $0$.  Additionally, we find that the quadruples $\{3,5,2,7\}$, $\{3,5,2,6\}$, and $\{5,2,4,6\}$ each contribute $-1$.  So, the final Casson count is $-1$, as desired. The above techiques yields a simple lower bound for the übercrossing number of Corollary <ref>:

\[
|c_2(K)|\leq {\textit{\"u}(K) -1 \choose 3}+{ \textit{\"u}(K) -1 \choose 4}.
\]

Indeed, the triple crossings contribute at most  ${\textit{\"u}(K) -1 \choose 3}$ to $c_2(K)$ and the quadruples at most ${ \textit{\"u}(K) -1 \choose 4}$.

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§ DIFFEOLOGY AND DIFFERENTIAL FORMS ON $\K$

A diffeology on a non-empty set $X$ is a set of parameterizations $\mathcal{D}=\{f:U\longrightarrow X\ |\ U\ \text{open in}\ \R^n,\ n=n(f)\}$ of $X$, called plots, such that three axioms are satisfied:

$(i)$ The set $\mathcal{D}$ contains all constant parameterizations $r\to x$ defined on $\R^n$, for all $x\in X$ and $n\geq 0$.

$(ii)$ Let $f : U \to X$ be a parameterization.  If for every point $r\in U$ there exists an open neighborhood $V$ of $r$ such that $f|_V$ belongs to $\mathcal{D}$, then $f$ belongs to $\mathcal{D}.$

$(iii)$ For every plot $f : U \to X$ of $\mathcal{D}$, every real domain $V$, and every $g\in C^\infty(V,U),$ $f\circ g$ belongs to $\mathcal{D}$.

A long curve is a smooth curve $\R\to\R^3$ which is the inclusion of the first coordinate axis except on a compact subset of $\R^3$. The space of long curves will be denoted by $\textrm{Map}_c(\R,\R^3)$, and the long knots $\K$ is a subspace of $\textrm{Map}_c(\R,\R^3)$.
A diffeology on $\K$ is induced from $\textrm{Map}_c(\R,\R^3)$ which carries a canonical diffeology; a map $f:U\to \textrm{Map}_c(\R,\R^3)$ from an open $U\subset\R^l$ is a plot if and only if the composite

\[
U\times \R\xrightarrow{\ f\times\textrm{id}\ } \textrm{Map}_c(\R,\R^3)\times \R\xrightarrow{\ \operatorname{ev}\ } \R^3
\]

is smooth.  In <cit.>, it is shown that this diffeology on $\textrm{Map}_c(\R,\R^3)$ is equivalent to the smooth diffeology on $\textrm{Map}_c(\R,\R^3)$, where plots are all smooth maps.
One can define forms on $\K$ as follows, a form $\omega$ is an assignment of a form $(\omega)_f$ on $U$ for each plot $f:U\longrightarrow \K$, such that, given a smooth map $h:V\longrightarrow U$ which induces a plot $f\circ h: V\longrightarrow \R^n$, we have: $(\omega)_{f\circ h}=h^\ast((\omega)_f).$

§ CONFIGURATION SPACE BUNDLES AND LOCAL COORDINATES

We now construct the bundles over which the configuration space integrals are defined and detail local coordinates for the fibers - compactified configuration spaces.  Similar constructions appear for closed knots in [8, 11] and [32], with a detailed explanation of coordinates given by Volić in [33].    Koytcheff, Munson, and Volić extend this framework to string links (and therefore long knots) in [24].  Additionally, in [31], Sinha gives an alternative and comprehensive look at the global and local coordinates for general compactified configuration spaces.

Section <ref> gives an overview of how this compactification of a configuration space is done, and provides a way of thinking about the boundary faces of the fiber.  Section <ref> gives an understanding about the local coordinates of a trivial configuration space bundle, and Section <ref> generalizes these to non-trivial configuration space bundles.  Both of these sections derive coordinates based off [33].

§.§ Fulton-MacPherson compactification

Let $M$ be a smooth manifold. Recall that
$\Cnf(M;n)$ denotes the $n$-fold product $M^n$ with all diagonals removed; i.e.,

\[
\Cnf(M;n)=\{(x_1,\dots,x_n)\in M^n|x_i\neq x_j \textrm{ for } i\neq j\}.
\]

The diagonals of $\Cnf(M;n)$ may be indexed by the subsets $S \subseteq \{1,...,n\}$ of cardinality at least $2$; let $\Delta_S$ denote the diagonal in $M^n$ where $x_i = x_j$ for all $i,j \in S$.  The blowup of $M^n$ along $\Delta_S$, denoted by $Bl(M^n,\Delta_S)$,  is a replacement of $\Delta_S$ in $M^n$  by the unit normal bundle. There is a natural embedding of  $M^n-\Delta_S$ to the interior of $Bl(M^n,\Delta_S)$, and a natural inclusion of $\Cnf(M;n)$ into $M^n$. Using the inclusion $\Cnf(M;n)\longrightarrow M^n-\Delta_S$ and the previously mentioned maps, one may  construct an embedding

\begin{equation}\label{eq:FM-map}
FM:\Cnf(M;n)\longrightarrow M^n\times \prod_{S\subset \{1,...,n\}, |S|\geq 2} Bl(M^n,\Delta_S).
\end{equation}

The closure of image of $\Cnf(M;n)$ under this map, denoted by $\Cnf[M;n]$, is known as the Fulton-MacPherson compactification of $\Cnf(M;n)$.
Note that $\Cnf[M;n]$ is the closure of the graph of

\[
\Cnf(M;n)\longrightarrow \prod_{S\subset \{1,...,n\}, |S|\geq 2} Bl(M^n,\Delta_S)
\]

in $M^n\times \prod_{S\subset \{1,...,n\}, |S|\geq 2} Bl(M^n,\Delta_S)$.
Some properties of $\Cnf[M;n]$ are listed below (proofs in [6, 31]):

$(i)$ the inclusion (<ref>) induces a homotopy equivalence between $\Cnf(M;n)$ and $\Cnf[M;n]$;

$(ii)$ $\Cnf[M;n]$ is a smooth $n$-dimensional manifold with corners, and it is compact if $M$ is compact;

$(iii)$ any embedding of $M$ in $N$ induces an embedding of $\Cnf[M;n]$ in $\Cnf[N;n]$.

We care particularly about the boundary of $\Cnf[M;n]$, which is a union of strata or faces. A stratum (face) corresponds to a subset of the $n$ configuration points degenerating (colliding or escaping to infinity). To simplify notation, assume there are $a$ degenerating points $x_i$, and these points are indexed by $\{1,2,...,a\}$. Either all $x_i$ degenerate to the same point, or disjoint subsets of $\{x_1,...,x_a\}$ degenerate to different points. Further, some points may degenerate faster than the others, meaning some points may collide in a first stage, and then with other points in a second stage, etc.  Therefore, the various ways that configuration points may degenerate can be efficiently tracked by nested subsets of $\{1,...,a\}$.

Specifically, every stratum, denote $S_{\mathcal{A}}$, can be labeled by $|\mathcal{A}|=j$ nested subsets $\mathcal{A}=\{A_1,\ldots, A_j\}$, where each of the subsets $A_p, A_q\subset \mathcal{A}$ has $|A_p|\geq2$, and either $A_p\cap A_q=\varnothing$, or $A_p\subseteq A_q$ (or vice versa).  As a direct consequence of coordinates description on $\Cnf[M;n]$ (see section <ref>), one obtains that the codimension of $S_{\mathcal{A}}$ is $j$.  In particular, faces of $\Cnf[M;n]$ of codimension $1$ correspond to some subset of the $n$ points either all colliding at the same time, or escaping to infinity at the same time.

In this paper, we consider configuration spaces of points in $\R$ and $\R^3$.

* When $M=\R$, the configuration space has $k!$ connected components.  One may then choose a single connected component to compactify and have as the domain of integration for a differential form.  We choose, without loss of generality, to use the connected component $\Cnf(\R;n)$, the component consisting of points $(x_1,\dots,x_n)$ such that $x_1<x_2<\dots<x_n$. The compactification of this component is sometimes denoted $\Cnf[\R;n]$.

$\Cnf[\R^n;n]$ is considered as the subspace of $\Cnf[S^n;n+1]$ where the last point is fixed at infinity.  ([24] Lemma 4.4) In this way, we can see that $\Cnf[\R;n]$ is compact.

§.§ Local Coordinates of $\Cnf[\R;n]\times\K$

Define $\Cnf[\R;n]\times\K$ to be the trivial bundle over $\K$ with projection map

\[
\pi:\Cnf[\R;n]\times\K\to\K
\]

which has as its fiber $\pi^{-1}(K)=\Cnf[\R;n]$ for $K\in \K$.  Additionally, define the following evaluation map:

\begin{equation}
\textrm{ev}:\Cnf[\R;n]\times\K\to\Cnf[\R^3,n],\\
\end{equation}

which is the extension of the evaluation of $n$ points in $\Cnf(\R;n)$ at $K\in\K$:

\[
\begin{split}
\Cnf(\R;n)\times\K&\to\Cnf(\R^3;n)\\
\end{split}
\]

On the interior of $\Cnf[\R;n]$, local coordinates are given by $(x_1,\dots,x_n)\in\Cnf(\R;n)$.  Composition of these coordinates with the evaluation map simply gives the evaluation map itself.

Next, we look to the boundary of $\Cnf[\R;n]$.  In particular, note that within each connected component of $\Cnf(\R;n)$, there is a total ordering on the $n$-coordinates. This ordering of coordinates fixes a choice of orientation for $\Cnf[\R;n]$ and its boundary faces. We focus on the connected component $C_n^\circ\subset\Cnf(\R;n)$ and the local coordinates of the boundary faces of $C_n$.

The codimension one faces of $\Cnf[\R;n]$ are given by:

* $S_{A\cup\{\infty\}}$ where $A\subseteq\{1,\dots,n\}$ and $A\neq\varnothing$ - these are faces where the points $x_i$ for $i\in A$ tend to $\infty$.

* $S_{A}$ where $A\subseteq\{1,\dots,n\}$ and $|A|\geq2$ - these are faces where the points $x_i$ for $i\in A$ collide away from $\infty$.

Following both [8] and [33], we label these faces in the following way.  A codimension one faces of $\Cnf[\R;n]$ (or for that matter, $\Cnf[\K,\R^3;n,s]$) is called:

* principal if exactly two points collide,

* hidden if more than two, but not all, points collide,

* anomalous if all points collide,

* a face at infinity if one or more points tend to infinity.

Since $x_1< x_2<\dots< x_n$ in $C_n^\circ$, if a face $S_A$ has $i,j\in A$ with $i< j$, then all $k$ such that $i\leq k\leq j$ are also in $A$.  I.e., if two points $x_i$ and $x_j$ collide (or tend to infinity), so will all points in between them.

First consider $A$ where $1\in A$; then $S_A$ is the face where the first $|A|=a$ points on the knot collide away from $\infty$.  Local coordinates for an open neighborhood of $S_A$ are given by

\[
\begin{split}
\hat{\phi}_A:\Cnf(\R;a)\times\Cnf(\R;n-a+1)\times[0,\infty)&\to\Cnf[\R;n]\\
(u_1,\dots,u_a,x_1,\dots,x_{n-a+1},r)&\to  (x_1+ru_1,\dots,x_1+ru_a,x_2,\dots, x_{n-a+1}).
\end{split}
\]

such that

\[
(1)\  x_i\neq x_j,\,  i\neq j,\qquad
(2)\ \displaystyle\sum_{i=1}^a u_i=0,\qquad
(3)\ \displaystyle\sum_{i=1}^a |u_i|^2=1.
\]

These first condition is required so that the image of $\phi_A$ is in the open neighborhood around $S_A$, the second two are required to make $\phi_A$ a diffeomorphism - the dimension of the domain of $\phi_A$ is $n+2$, while the dimension of $\Cnf[\R;n]$ is $n$.  Therefore, we need to restrict the domain of $\phi_A$ to some codimension $2$ subspace.  Points on the face $S_A$ are those for which $r=0$.
Composing these coordinates with the evaluation map gives $\phi_A=\textrm{ev}\circ\hat{\phi}_A$ with

\[
\begin{split}
\phi_A:\Cnf(\R;a)\times\Cnf(\R;n-a+1)\times[0,\infty)\times\K&\to\Cnf[\R^3;n]\\
\phi_A(u,x,r,K)&=\begin{cases}K(x_1+ru_i)&\textrm{ if }i\in A\\ K(x_i)&\textrm{ if }i\notin A.\end{cases}.
\end{split}
\]

* To derive local coordinates for $S_{A\cup\{\infty\}}$ we may use the same map $\phi_A$; points on $S_{A\cup\{\infty\}}$ are obtained by taking the limit as $r\to\infty$..

* When $1\notin A$, suppose that $i$ is the smallest index where $i\in A$, and $|A|=a$.  Then local coordinates for $S_{A}$ and $S_{A\cup\{\infty\}}$ will be given by

\[\phi_A=s^{i-1}\circ\phi_{A'}\circ s^{-(i-1)},\]

where $A'=\{1,\dots,a\}$, and $s$ is a map which cyclicly permutes coordinates.  It is then the case that  $\phi_A^\ast\sigma=(-1)^{(n+1)(i-1)}(-1)^{(n-a+1)(i-1)}\phi_{A'}^\ast\sigma.$ (See [8] for more specifics on this map $s$.)

* In a more general setting of trivalent/arrow diagrams, trivial configuration space bundles are associated to diagrams with $n$ segment and $0$ free vertices.

§.§ Local Coordinates of $\Cnf[\K,\R^3;n,s]$

Define the following projection map between configuration spaces:

\begin{equation}
\textrm{p}:\Cnf[\R^3;n+s]\to\Cnf[\R^3;n],
\end{equation}

which is forgetting the last $s$ points of the configuration. From this projection map, we create the following pullback bundle.

The fiber bundle $\Cnf[\mathcal{K},\R^3;n,s]$ is given as the pullback bundle over $p$ in the following diagram

\[
\xymatrix{
\Cnf[\mathcal{K},\R^3;n,s] \ar[rr] \ar^-{}[rr] \ar@{->}_-{p'}[d]  & & \Cnf[\R^3;n+s] \ar^-{p}[d] \\
\Cnf[\R;n]\times \mathcal{K}  \ar^-{\ev}_-{\ }[rr] & &  \Cnf[\R^3;n].
\]

We define $\Cnf(\K,\R^3;n,s)$ analogously, and have the following explicit description:

\[
\begin{split}
\Cnf(\mathcal{K},\R^3;n,s)& = \{ (x_1,\ldots,x_n,K,\mathbf{x}_{1},\ldots, \mathbf{x}_{n+s})\in \Cnf(\R;n)\times \mathcal{K}\times \Cnf(\R^3;n+s)|\\
& \qquad K(x_i)=\mathbf{x}_i,\ 1\leq i\leq n\}.
\end{split}
\]

$\Cnf[\mathcal{K},\R^3;n,s]$ is a smooth fiber bundle whose fiber $\pi^{-1}(K)$ is a smooth, finite dimensional manifold with corners (see <cit.> or <cit.>), which is independent of $K$, and depends only on $n$ and $s$.

For a given knot $K\in\K$, $\pi^{-1}(K)$, can be thought of as a $3s+n$ dimensional configuration space in the following way.  For $p\in$int$(\pi^{-1}(K))$, the first $n$ coordinates correspond to $n$ distinct points on $K$, and the last $s$ coordinates can be anything in $\R^3-\{K(x_1),\ldots,K(x_n)\}$.  We refer to the first $n$ coordinates of $p$ as “points on the knot" and the last $s$ as “points off the knot", although they can also lie on $K$, as long as they are distinct from the points $\{K(x_1),\ldots,K(x_n)\}$ and each other.  For points on the boundary of $\pi^{-1}(K)$, the first $n$ coordinates lie on $K$ and the last $s$ coordinates of $p$ can be anything in $\R^3$.

The codimension one faces of $\pi^{-1}(K)$ are given by:

* $S_{A\cup\{\infty\}}$ where $A\subseteq\{1,\dots,n\}$ and $A\neq\varnothing$ - these are faces where the points on the knot $x_i$ for $i\in A$ tend to $\infty$, and points off the knot remain distinct.

* $S_{A}$ where $A\subseteq\{1,\dots,n\}$ and $|A|\geq2$ - these are faces where the points on the knot $x_i$ for $i\in A$ collide away from $\infty$ and points off the knot remain distinct.

* $S_{B\cup\{\infty\}}$ where $B\subseteq\{n+1,\dots,n+s\}$ and $B\neq\varnothing$ - these are faces where the points off the knot $\mathbf{x}_i$ for $i\in B$ tend to $\infty$, and points on the knot remain distinct.

* $S_{B}$ where $B\subseteq\{n+1,\dots,n+s\}$ and $|B|\geq2$ - these are faces where the points off the knot $\mathbf{x}_i$ for $i\in B$ collide away from $\infty$ and points on the knot remain distinct.

* $S_{A\cup B\cup\{\infty\}}$ where $A\subseteq\{1,\dots,n\}, B\subseteq\{n+1,\dots,n+s\}$ and $A,B\neq\varnothing$ - these are faces where the points $\mathbf{x}_i$ for $i\in B$ and $K(x_j)$ for $j\in A$ tend to $\infty$.[Since $K$ is a long knot and is the inclusion of the $x$ axis except on some compact subset, for $x_i$ large enough, $K(x_i)=(x_i,0,0)$.]

* $S_{A\cup B}$ where $A\subseteq\{1,\dots,n\}, B\subseteq\{n+1,\dots,n+s\}$ and $A,B\neq\varnothing$ - these are faces where the points $\mathbf{x}_i$ for $i\in B$ and $K(x_j)$ for $j\in A$ collide away from infinity.

One may divide the local coordinates of these faces into cases, based on if $A=\varnothing$, $B=\varnothing$, or both $A,B\neq\varnothing$: these are cases when only points off the knot collide , only points on the knot collide, or points off and on the knot collide respectively.   We provide local coordinates for this last case; for more details on the other cases, see [33], [10].

To derive coordinates in the case that $A\neq\varnothing$ and $B\neq\varnothing$, let us consider the $a$ points colliding on the knot indexed (for convenience) by $A=\{1,\ldots,a\}$, and the $b$ points colliding off the knot indexed by $B=\{n+1,\ldots, n+b\}$. Then $S_{A\cup B}$ is given by

\[
\begin{split}
S_{A\cup B} &=\{(x_1,\dots,x_n,K(\mathbf{z}),\mathbf{x}_1,\dots,\mathbf{x}_{n+s})\in \Cnf[\mathcal{K},\R^3;n,s] |\\
&\qquad K(x_i,\mathbf{z})=\mathbf{x}_i,\ 1\leq i\leq n,\\
&\qquad  x_1=x_j, \  1\leq j\leq a,\\
&\qquad  K(x_1,\mathbf{z})=\mathbf{x}_{l}  \ n+1\leq l\leq n+b
\}.
\end{split}
\]

On this face, $a$ points on the knot and $b$ points off of the knot are colliding with $K(x_1)$, and no subset of these colliding points converge before they all do.  On the face itself, coordinates should record $n-a+1$ distinct points on the knot, $s-b$ points off of the knot, and directions of convergence for all $a+b$ colliding points.  The $a$ points on the knot are restricted in their direction of convergence; this direction must be some distinct multiple of $K'(x_1)$.  The $b$ points off of the knot may converge from any direction, but each of these $b$ directions must be distinct from each other and the directions of convergence of the $a$ points on the knot. Therefore, to parameterize a neighborhood of the face $S_{A\cup B}$, we set up a type of spherical coordinates to encode all this information in the limit.

For a given $K$, let

\[
U_{r\geq 0}=\Cnf(\R;a)\times\Cnf(\R^3,b)\times \Cnf(\R;n-a+1)\times \Cnf(\R^3;s-b)\times  [0,\infty)
\]

where $ (u,\mathbf{u},x,\mathbf{x},r)\in U_{r\geq 0}$ has the form

\[
\begin{split}
(u,\mathbf{u},x,\mathbf{x},r) & =(u_1,\ldots, u_a, \mathbf{u}_{a+1},\ldots, \mathbf{u}_{a+b},x_1,\ldots, x_{n-a+1},  \mathbf{x}_{n-a+2},\ldots, \mathbf{x}_{n+s-a-b+1}, r).
\end{split}
\]

We define the following map parameterizing the neighborhood of a face $S_{A\cup B}$ in the interior of $\Cnf[\K,$ $\R^3;n,s]$, for a given plot $K(\mathbf{z})\in\K$ parameterized by $\mathbf{z}\in U$:

\[
\begin{split}
\phi_{A\cup B}:  U_{r\geq0}\times U&\longrightarrow  \Cnf(\K,\R^3;n,s)\subset \Cnf(\R;n)\times\K\times\Cnf(\R^3;n+s)\\
\phi_{A\cup B}(u,\mathbf{u},x,\mathbf{x},r,\mathbf{z}) &=(x_1+r u_1,\dots,x_1+r u_a,x_2,\dots,x_{n-a+1},\\
&\qquad K(\mathbf{z}),  K(x_1+r u_1,\mathbf{z}),\dots,K(x_1+r u_a,\mathbf{z}),\\
&\qquad K(x_2,\mathbf{z}),\dots,K(x_{n-a+1},\mathbf{z}),\\
&\qquad K(x_1,\mathbf{z})+r\mathbf{u}_{a+1},\dots,K(x_1,\mathbf{z})+r\mathbf{u}_{a+b} ,\\
&\qquad \mathbf{x}_{n-a+2},\ldots, \mathbf{x}_{n+s-a-b+1})
\end{split}
\]

such that

* $ K'(x_1,\mathbf{z})u_i\neq\mathbf{u_j}  \qquad i\leq a,\ \  j>a$

* $K(x_i,\mathbf{z})\neq\mathbf{x}_j \qquad 1\leq i \leq n-a+1, \ \ n-a+2\leq j\leq n+s-a-b+1$

* $\displaystyle\sum_{i=1}^a|K'(x_1,\mathbf{z})|^2u_i^2+\displaystyle\sum_{i=a+1}^{a+b}|\mathbf{u}_i|^2=1$

* $  \displaystyle\sum_{i=1}^a u_i+\displaystyle\sum_{i=a+1}^{a+b}\frac{\langle K'(x_1,\mathbf{z}),\mathbf{u}_i\rangle}{|K'(x_1,\mathbf{z})|^2}=0.$

Note that $u\in\Cnf(\R;a)$ prevents two points on the knot from colliding before the limit, and $\mathbf{u}\in\Cnf(\R^3;b)$  prevents two points off the knot from colliding before the limit.  Condition <ref> prevents a collision between one point off the knot and one point on the knot, before they all come together.  This ensures that each component of the vector of limiting directions is distinct, as the limiting directions between colliding points are represented by

\[
(K'(x_1,\mathbf{z})u_1,\dots K'(x_1,\mathbf{z})u_a, \mathbf{u}_{a+1}\dots\mathbf{u}_{a+b})
\]

in the unit normal bundle to $S_{A\cup B}$.  Condition  <ref> ensures that the image of $\phi_{A\cup B}$ is in the pullback bundle.  Conditions  <ref> and  <ref> ensure that the vector of limiting directions is in the unit sphere bundle of the normal bundle.

The parametrization $\overline{\phi}_{A\cup B}:U_{r\geq 0}\times U\longrightarrow \Cnf[\mathcal{K},\R^3;n,s]$ on these coordinates is the continuous extension of $\phi_{A\cup B}$ to the compactification $\Cnf[\K,\R^3;n,s]$  after composing with the $FM$ map of (<ref>) on both $\Cnf(\R^3;n+s)$  and $\Cnf(\R;n)$.   (for the sake of simplified notation, we refer to the map $\overline{\phi}_{A\cup B}$ as simply ${\phi}_{A\cup B}$.)

This is the case where the colliding points were the first $a$ points on the knot, and the first $b$ points off the knot.  However, a codimension one face may have any collection of $a$ adjacent segment vertices and any collection of $b$ points off the knot colliding.  To provide coordinates for these faces, the process is the same; one may choose the smallest point on the knot $x_l$, and write the above coordinates colliding onto $K(x_l,\mathbf{z})$.  Writing it down and keeping track of the indices becomes slightly more complicated.