# FRAMED KNOTOIDS AND THEIR QUANTUM INVARIANTS

WOUT MOLTMAKER

###### Abstract

We modify the definition of spherical knotoids to include a framing, in
analogy to framed knots, and define a further modification that includes a
secondary ‘coframing’ to obtain ‘biframed’ knotoids. We exhibit topological
spaces whose ambient isotopy classes are in one-to-one correspondence with
framed and biframed knotoids respectively. We then show how framed and
biframed knotoids allow us to generalize quantum knot invariants to a knotoid
setting, leading to the construction of general Reshetikhin-Turaev type
biframed knotoid invariants.

##  1 Introduction and Motivation

Knotoids were introduced by V. Turaev in [15] as a generalization of knots.
Intuitively, knotoids are knot diagrams that have open ends; see Figure 2.1
below. Unlike long knots [4, Ch. 1], these end-points are allowed to lie in
the interior of the diagram. To prevent all knotoid diagrams from being
trivial, the diagrammatic moves in Figure 1.1 are explicitly forbidden.

Figure 1.1: The forbidden knotoid diagram moves.

Turaev showed that such diagrams are in correspondence with so-called ‘simple
theta-curves’, in the same way that knot diagrams correspond to knots. In this
sense we say that simple theta-curves are the ‘geometric realization’ of
knotoid diagrams.

Knotoids are a natural object to use when studying the knottedness of open-
knotted protein chains. This is done in e.g. [7] where the knottedness of a
protein is analysed by projecting it onto planes in many possible directions
and identifying the resulting knotoids. The result is a map from a
discretization of $S^{2}$ to the set of all knotoids, which encodes much
information about the shape of the protein.

To make this application of knotoids viable for more complex proteins, it is
helpful to have a classification of knotoids. The classification of knotoids
on the sphere with up to 6 crossings is complete [6], and makes use of strong
knotoid invariants such as the arrow polynomial from [8] and the double
branched cover from [2]. For further progress a larger selection of knotoid
invariants is needed. Extending this selection is the primary goal of of this
paper.

In this paper the notion of a knotoid diagram is generalized to include a
framing, in analogy to framed knots. It is generally easier to construct
invariants of framed knots since framed knot diagrams have fewer allowed
Reidemeister moves than knot diagrams. This same principle motivates the
introduction of framed knotoids to facilitate the production of knotoid
invariants. Along with definitions of several notions of framed knotoid
diagrams we will also give geometric realizations of these types of diagrams.
This helps to keep the theory grounded in topology, before allowing algebraic
and diagrammatic manipulations to take over when we move on to constructing
framed knotoid invariants.

This construction of framed knotoid invariants leads to the construction of
general Reshetikhin-Turaev type quantum invariants of knotoids. This is
related to the very recent work by Gügümcü and Kauffman [9] on quantum knotoid
invariants. Our work in this paper is independent from theirs, but there are
similarities between our approach and theirs: e.g. what we shall call our
‘coframing’ is analogous to their ‘rotation number’. Our approach will add
some topological motivation to the ideas developed there, and as a result has
a focus on spherical knotoids rather than planar ones.

##  2 Definitions and Geometric Realizations

In this section we briefly recall the basic results on knotoids and simple
theta-curves, both due to Turaev [15]. Afterwards we introduce framed and
biframed versions of both knotoids and simple theta-curves.

###  2.1 Knotoids and Simple Theta-curves

######  Definition 2.1.

[15] Let $\Sigma$ be a surface. A knotoid diagram in $\Sigma$ is a smooth
immersion $\phi:[0,1]\hookrightarrow\Sigma^{\circ}$ into the interior of
$\Sigma$, whose only singularities are transversal double points endowed with
over/undercrossing data. Knotoid diagrams have a natural orientation, namely
from the leg $\phi(0)$ to the head $\phi(1)$. We will not always label the leg
and head or depict knotoids as oriented, but this orientation is always
implicit. Two knotoid diagrams are equivalent if they can be related by a
sequence of isotopies of $\Sigma$ and applications of the Reidemeister moves
$R1,R2,R3$ familiar for knot diagrams. Knotoids are the equivalence classes of
knotoid diagrams.

Applications of the Reidemeister moves are not allowed to involve the end-
points, and indeed the diagrammatic moves in Figure 1.1 are expressly
forbidden.

Figure 2.1: Examples of knotoid diagrams on $\Sigma=\mathbb{R}^{2}$.

The following is immediate from noting that $S^{2}$ is isomorphic to the one-
point compactification of $\mathbb{R}^{2}$.

######  Remark 2.2.

Knotoids in $\Sigma=S^{2}$ are equivalent to knotoids in
$\Sigma=\mathbb{R}^{2}$ modulo the spherical move $R4$ depicted in Figure 2.2.

Figure 2.2: The spherical move on a knotoid diagram, where $K$ denotes the
rest of a fixed knotoid diagram.

Note that while this move needs to be imposed on knotoid diagrams in
$\mathbb{R}^{2}$ and may change the equivalence class of a knotoid diagram in
$\mathbb{R}^{2}$, it is a surface isotopy for knotoids in $S^{2}$ given by
sweeping a strand external to the diagram along the back of $S^{2}$.
Nevertheless, it will be helpful to distinguish this move from other
isotopies, so that we are free to think of knotoids in $S^{2}$ as lying in
$\mathbb{R}^{2}$.

We call a knotoid in $S^{2}$ spherical. In this paper all knotoids are assumed
to be spherical unless stated otherwise.

######  Remark 2.3.

In the remainder of this section we will consider several kinds of embeddings
of topological spaces in $S^{3}$, related to ‘theta-curves’ defined below. All
of these topological spaces will be locally homeomorphic to $\mathbb{R}$,
$\mathbb{R}^{2}$, or $\mathbb{R}\times[0,\infty)$ except at two points or
arcs. All the embeddings that we will consider in what follows will be
implicitly assumed to be smooth everywhere except for these points or arcs,
where smoothness is not defined.

######  Definition 2.4.

[15] A theta-curve $\theta$ is an embedding into $S^{3}$ of the graph
$\Theta$, which consists of two vertices $v_{0},v_{1}$ and three edges
$e_{-},e_{0},e_{+}$ joining them. We consider such embeddings up to ambient
isotopies of $S^{3}$ that preserve the labels of the vertices and edges. If
$\theta$ is a theta-curve, then omitting any edge from $\Theta$ results in an
embedding $S^{1}\hookrightarrow S^{3}$, i.e. a knot. A theta-curve $\theta$ is
said to be simple if the image of $e_{-}\cup e_{+}$ is the unknot.

The reason for introducing simple theta-curves is the following geometric
realization result, also due to Turaev [15]:

######  Theorem 2.5.

There is a bijection between spherical knotoids and label-preserving ambient
isotopy classes of simple theta-curves. We say that $\Theta$ is therefore the
geometric realization of knotoids.

For a detailed proof, see [15, Sec. 6]. For us the important parts of the
proof are the explicit correspondence, and the notion of a standard theta-
curve. Intuitively a theta-curve is standard if $e_{+}\cup e_{-}$ is
manifestly the unknot:

######  Definition 2.6.

A simple theta-curve $\theta$ is standard if the vertices of $\theta$ lie in
$\mathbb{R}^{2}\times\\{0\\}\subseteq
S^{3}\cong\mathbb{R}^{3}\cup\\{\infty\\}$, $\theta(e_{+}),\theta(e_{-})$ lie
in the upper- and lower half-plane of $\mathbb{R}^{2}\times\\{0\\}$
respectively, and $\theta(e_{+}),\theta(e_{-})$ project bijectively to the
same arc $a\subseteq\mathbb{R}^{2}\times\\{0\\}$ that connects the vertices of
$\theta$.

Turaev shows that any simple theta-curve is ambient isotopic to a standard
one, and that if $\theta,\theta^{\prime}$ are ambient isotopic standard simple
theta-curves then they are ambient isotopic within the class of standard
simple theta-curves [15]. For standard simple theta-curves the bijection of
Theorem 2.5 is depicted in Figure 2.3.

Figure 2.3: The bijection of Theorem 2.5. A standard theta-curve produces a
knotoid by moving $\theta(e_{0})$ onto a neighborhood of
$\mathbb{R}^{2}\times\\{0\\}$ and taking a vertical planar projection. A
knotoid conversely produces a theta-curve by considering a knotoid diagram as
a knotted arc lying in a neighbourhood of $\mathbb{R}^{2}\times\\{0\\}$ and
tying arcs $e_{\pm}$ to its end-points.

###  2.2 Framed and Biframed Knotoids

The aim of this section is to give reasonable definitions of what it means for
a knotoid to be framed. First recall the main results on framed knots from
[5]: A framed knot $K$ is a knot with a transversal, smooth, everywhere
nonzero vector field. The framing of $K$ is then defined to be the associated
element in $\pi_{1}(SO(2))\cong\mathbb{Z}$. There is an equivalence between
framed knots and tangled ribbons in $S^{3}$, so framed knots can equivalently
be characterized as embeddings of the annulus into $S^{3}$.

Any knot diagram induces a blackboard framing on the corresponding knot, with
framing integer given by the writhe of the diagram. Letting $R1^{\prime}$
denote the weakened first Reidemeister move (see Figure 2.4), framed knots up
to equivalence are in bijection with knot diagrams up to $R1^{\prime},R2,R3$.
Such knot diagrams are referred to as framed knot diagrams. Details can be
found in [5].

Figure 2.4: The weakened first Reidemeister move, $R1^{\prime}$.

####  2.2.1 Framed Knotoids

Given that framed knot diagrams are simply knot diagrams up to the moves
$R1^{\prime},R2,R3$, the definition of framed knotoids is natural:

######  Definition 2.7.

A framed knotoid is an equivalence class of knotoid diagrams under the
equivalence generated by ambient isotopies and $R1^{\prime},R2,R3$. A diagram
representing a framed knotoid is also called a framed knotoid diagram.

As the terminology suggests, framed knotoid diagrams are just knotoid diagrams
with a ‘framing integer’ attached. Our immediate goal is to prove this. To
extract an integer from a given diagram $K$ in a canonical way we proceed in
analogy with framed knots:

######  Definition 2.8.

Let $K$ be a framed knotoid diagram. We define the blackboard framing of $K$
to be the writhe of $K$, i.e. the sum of the signs of all crossings of $K$.
Here the sign of a crossing is defined analogously as for knot diagrams, see
[5].

The blackboard framing is an integer that is also well-defined for the framed
knotoid that $K$ represents. This last statement follows from invariance of
the writhe under $R1^{\prime},R2,R3$. To justify the use of the term
‘blackboard framing’ for the writhe of a knotoid in analogy with framed knots,
we need to interpret the framed knotoid as an object in $S^{3}$ with an
associated canonical transversal vector field. We do so after introducing
framed theta-curves in the next subsection. For now it suffices to work with
the combinatorially defined writhe, without reference to a geometric
realization. Using the writhe, we conclude the following lemma:

######  Lemma 2.9.

There is a bijection

$\displaystyle\\{\text{Framed knotoids}\\}$
$\displaystyle\leftrightarrow\\{\text{Knotoids}\\}\times\mathbb{Z}$ (2.1)
$\displaystyle K$ $\displaystyle\mapsto\left(K,\text{writhe}(K)\right).$

###### Proof.

First note that surjectivity is clear. Indeed, given a knotoid diagram $K$ we
can add $R1$ loops to adjust the framing of $K$ at will without altering the
underlying knotoid. (By an $R1$ loop we mean a loop like those pictured in
Figure 2.4.) For injectivity, suppose that $K,K^{\prime}$ are equivalent
knotoid diagrams with equal writhe. Then we must show that $K,K^{\prime}$ are
equivalent as framed knotoid diagrams. To see this, first note that positive
and negative $R1$ loops can be cancelled against each other via a combination
of $R2$ and $R3$ moves. See Figure 2.5.

Figure 2.5: Cancellation of opposite $R1$ loops.

Since $K,K^{\prime}$ are equivalent as knotoids there is a sequence of
isotopies of $S^{2}$ and Reidemeister moves that turns $K^{\prime}$ into $K$.
By the above, we can modify this sequence of moves into a sequence of framed
knotoid diagram moves. Namely we replace every deletion of an $R1$ loop by the
sequence of isotopies of $S^{2}$ and $R2,R3$ moves that brings this $R1$ loop
to a neighbourhood of the head of $K^{\prime}$, and every creation of an $R1$
loop by the creation of that loop and its opposite, followed by moving this
opposite to the head of $K^{\prime}$. The result is the diagram $K$, but with
a finite number of $R1$ loops concentrated in a neighbourhood of the head.
Since $K^{\prime}$ and $K$ have equal writhe, the sum of the signs of these
loops is $0$. Thus we can apply the moves in Figure 2.5 finitely many times to
remove all these loops, ending up with a diagram of $K$. ∎

####  2.2.2 Framed Simple Theta-curves

Our next goal is to find a geometric realization of framed knotoids, and to
interpret their writhe in terms of the framing of this geometric realization.

This means that we want topological spaces whose ambient isotopy classes in
$S^{3}$ are in bijection with framed knotoids. Recall that framed knots are
equivalent to tangled ribbons in $S^{3}$, and hence the geometric realization
of framed knot diagrams is the annulus. Also recall the sketch proof of
Theorem 2.5, which implies that knotoids correspond to the edge $e_{0}$ of a
simple theta-curve. From these two facts it is reasonable to expect that
framed knotoids correspond to simple theta-curves in $S^{3}$, with $e_{0}$
thickened into a ribbon. The associated geometric realization is pictured in
Figure 2.6.

Figure 2.6: A simple framed theta-curve.

######  Definition 2.10.

Let $\overline{\Theta}$ denote the space $\Theta$ with the middle edge $e_{0}$
thickened into a 2-dimensional ribbon $\overline{e}_{0}$ attached along
neighbourhoods of $v_{0},v_{1}$ as depicted in Figure 2.6. A framed theta-
curve is an embedding $\overline{\Theta}\hookrightarrow S^{3}$. We consider
such embeddings up to label-preserving ambient isotopies of $S^{3}$.

We say that a framed theta-curve is simple if it has integer framing and the
image of $e_{+}\cup e_{-}$ is the unknot, the former meaning that if we put
$\theta$ in a form such that $e_{+}\cup e_{-}$ is manifestly unknotted then
the same side of $\overline{e}_{0}$ must face ‘outwards’ at $v_{0}$ and
$v_{1}$. In other words, it means that we want $\theta(\overline{e}_{0})$ to
make an integer number of turns through $S^{3}$, in some sense; see Figure
2.6.

To make this notion of ‘turns’ more precise we define standard framed simple
theta-curves and their framing integers:

######  Definition 2.11.

Let $\mathbb{R}^{2}$ denote the plane
$\mathbb{R}^{2}\times\\{0\\}\subseteq\mathbb{R}^{3}\subseteq S^{3}$. In
analogy to the proof of Theorem 2.5 we say that a framed simple theta-curve
$\theta$ is standard if the same assumptions on $\theta(e_{+}),\theta(e_{-})$
as for standard theta-curves hold, and the neighbourhoods of $\theta(e_{+}\cup
e_{-})$ where $\theta(\overline{e}_{0})$ is attached are parallel straight
lines in $\mathbb{R}^{2}$. In particular the example in Figure 2.6 is
standard.

Let $e_{0}$ be one of the lengths of the boundary of
$\theta(\overline{e}_{0})$. The direction into the interior of
$\overline{e}_{0}$ that is transversal to $e_{0}$ defines a path in $SO(2)$
when considered along all of $\theta(\overline{e}_{0})$. In the case of a
standard simple framed theta-curve this path is a loop by assumption. The
corresponding element of $\pi_{1}(SO(2))\cong\mathbb{Z}$ is defined to be the
framing integer $\text{fr}(\theta)$ of $\theta$.

######  Remark 2.12.

Note that any framed simple theta-curve can be brought into standard form.
Indeed, it suffices to extend the same moves that bring a simple theta-curve
into standard form to a framed simple theta-curve, and then to adjust regular
neighbourhoods of the end-points to be vertical. (For the definition of a
regular neighbourhood of an end-point, see [15]. Here we assumed without loss
of generality that the attachments of $\overline{e}_{0}$ lie inside such a
neighbourhood.) This works because any move on theta-curves can easily be
extended to a move of framed theta-curves, up to deformations that shrink a
portion of the ribbon $\overline{e}_{0}$. We will see in subsection 2.2.3 that
for biframed theta-curves this is not quite the case.

It is clear that the framing integer is independent of the standard form
produced in this way. Indeed, the relative twisting between the transversal
vectors to $e_{0}$ into $\overline{e}_{0}$ at $v_{0},v_{1}$ when transforming
a theta-curve into standard form is clearly independent of the standard form
chosen.

Remark 2.12 allows us to define the framing integer of any simple theta-curve
$\theta$, namely as the framing integer of any standard theta-curve equivalent
to $\theta$. Using this we can conclude the following:

######  Lemma 2.13.

There is a bijection

$\displaystyle\\{\text{Framed simple theta-curves}\\}$
$\displaystyle\leftrightarrow\\{\text{Simple theta-curves}\\}\times\mathbb{Z}$
(2.2) $\displaystyle\theta$
$\displaystyle\mapsto\left(\theta^{\circ},\text{fr}(\theta)\right),$

where $\theta^{\circ}$ is the simple theta-curve obtained from the projection
$\overline{e}_{0}\twoheadrightarrow e_{0}$ of a ribbon onto one of its
$1$-dimensional boundary lengths (which is a smooth arc).

###### Proof.

This proof is analogous to that of Lemma 2.9: surjectivity is clear. For
injectivity suppose that $\theta_{1},\theta_{2}$ have equal framing and that
$\theta_{1}^{\circ}\simeq\theta_{2}^{\circ}$. Then an ambient isotopy relating
$\theta_{1}^{\circ}$ to $\theta_{2}^{\circ}$ can be extended to an ambient
isotopy relating $\theta_{1}$ to a theta-curve $\theta_{3}$ such that
$\theta_{3}^{\circ}$ is equal to $\theta_{2}^{\circ}$. This extension may end
up adding extra twists (both clockwise and counter-clockwise) to $\theta_{3}$
that are not present in $\theta_{2}$. But since
$\text{fr}(\theta_{3})=\text{fr}(\theta_{1})=\text{fr}(\theta_{2})$ these
extra twists must cancel to give $\theta_{1}\simeq\theta_{3}\simeq\theta_{2}$,
proving injectivity. ∎

Combining bijection (2.2) with bijection (2.1) and Theorem 2.5 gives a
sequence of bijections

$\displaystyle\\{\text{Framed knotoids}\\}$
$\displaystyle\xleftrightarrow{\text{writhe}}\\{\text{Knotoids}\\}\times\mathbb{Z}$
$\displaystyle\xleftrightarrow{\text{Thm
}\ref{thm:knotoidbijection}}\\{\text{Simple theta-curves}\\}\times\mathbb{Z}$
$\displaystyle\xleftrightarrow{\text{framing}}\\{\text{Framed simple theta-
curves}\\}.$ (2.3)

Thus we obtain the following corollary to Theorem 2.5:

######  Corollary 2.14.

The geometric realization of framed knotoids is $\overline{\Theta}$, i.e. the
equivalence classes of embeddings of $\overline{\Theta}$ are in bijective
correspondence with framed knotoids.

######  Remark 2.15.

A priori the bijection of Corollary 2.14 is not topological in nature: it
simply says that both framed knotoids and framed simple theta-curves
correspond to knotoids with an integer attached. As with framed knots, the
topological interpretation of Corollary 2.14 passes through the blackboard
framing.

Figure 2.7: The blackboard framing of a knotoid diagram.

Given a knotoid diagram $K$, let $K^{\prime}$ be given by a small shift
transversal to $K$; see Figure 2.7. Under the inverse of the projection
assignment from Theorem 2.5, this $K^{\prime}$ specifies a transversal vector
field on $e_{0}$ of the standard theta-curve corresponding to $K$. (As a
convention, we take this vector field to be upwards in
$\mathbb{R}^{2}\times\\{0\\}$ at $v_{0},v_{1}$.) In turn this specifies a
ribbon extension $\overline{e}_{0}$ to $e_{0}$. The corresponding framing is
defined to be the blackboard framing of $K$. This is a number equal to the
writhe of $K$. The proof of this is analogous to that for framed knots; see
[5].

In this way, the notions of framing for knotoids and theta-curves coincide
under the map $\\{\text{Knotoids}\\}\to\\{\text{Simple theta-curves}\\}$ from
Theorem 2.5.

Framed simple theta-curves, and hence framed knotoids, could be useful in
their own right for modelling DNA structures. Namely the framing can be used
to model the double helix structure of a DNA strand more accurately. A
consequence is that framed knotoid models of DNA can detect features such as
supercoiling [3], which is undetectable by knotoid models due to the $R1$
relation. For such modelling purposes, it is natural to ask whether knotoids
can be generalized to encode non-integer framing. Clearly this can be done for
framed simple theta-curves, but there is no simple way to translate this to
knotoid diagrams.

Defining a notion of half-integer framing that is easy to recognize in
diagrams is one of the motivations for introducing ‘biframed’ knotoids and
theta-curves in the next subsections.

####  2.2.3 Biframed Simple Theta-curves

To define biframed knotoids we will work backwards; we start by defining the
geometric realization and its embeddings, and consider the associated diagrams
in the next subsection. The idea of a biframed theta-curve is to take a framed
theta-curve, and add a secondary coframing by allowing the outer ring
$e_{+}\cup e_{-}$ to carry framing information. The associated geometric
realization is depicted in Figure 2.8.

Figure 2.8: A (simple) biframed theta-curve.

######  Definition 2.16.

Let $\lvert\Theta\rvert$ denote the space depicted on the left in Figure 2.8,
i.e. the space $\Theta$ with $\\{e_{-},e_{0},e_{+}\\}$ thickened into ribbons
$\\{\overline{e}_{-},\overline{e}_{0},\overline{e}_{+}\\}$ and with
$\overline{e}_{0}$ attached at attaching arcs $a_{0},a_{1}$ along the width of
$\overline{e}_{-}\cup\overline{e}_{+}$. This means that if we note that
$\overline{e}_{+}\cup\overline{e}_{-}\cong S^{1}\times[0,1]$ then the arcs
$a_{i}$ are equal to $p_{i}\times[0,1]$ for some $p_{i}\in S^{1}$ and
$i\in\\{0,1\\}$. A biframed theta-curve is an embedding
$\theta:\lvert\Theta\rvert\hookrightarrow S^{3}$. We consider such embeddings
up to label-preserving ambient isotopies of $S^{3}$.

To justify the terminology ‘biframed’, we give a way of assigning two numbers
to (a certain class of) biframed theta-curves. One of these is analogous to
the framing of a framed theta-curve. The other is derived from the fact that
for a biframed theta-curve $\overline{e}_{-}\cup\overline{e}_{+}$ is also a
ribbon, and shall be referred to as the ‘coframing’ of a biframed theta-curve.
As for framed theta-curves, to facilitate the definition of these numbers we
first define a class of ‘standard’ theta-curves.

######  Definition 2.17.

A biframed theta-curve $\theta$ is said to be standard if the following
conditions hold:

  * •

The annulus $\theta(\overline{e}_{-}\cup\overline{e}_{+})$ is the unframed
(i.e. framing $0$) unknot, and $\theta(a_{i})$ lies in the horizontal plane
$\mathbb{R}^{2}\times\\{0\\}\subseteq S^{3}$ for $i\in\\{0,1\\}$.

  * •

The annulus $\theta(\overline{e}_{-}\cup\overline{e}_{+})$ is in manifestly
unframed form, i.e. of the form $S^{1}\times[-\epsilon,\epsilon]$ with
$S^{1}\subseteq\mathbb{R}^{2}\times\\{0\\}\subseteq S^{3}$.

  * •

Let $e_{0}$ denote one of the lengths of the boundary of
$\theta(\overline{e}_{0})$. Some neighbourhoods of the attaching arcs of
$\theta(\overline{e}_{0})$ to $\theta(\overline{e}_{-}\cup\overline{e}_{+})$
lie in a single plane $P$, and the tangent of $e_{0}$ is perpendicular to
$\theta(\overline{e}_{-}\cup\overline{e}_{+})$ at both attaching points.

  * •

Away from the attaching arcs, $\theta(\overline{e}_{0})$ does not intersect
the vertical bars perpendicular to $P$ that are depicted in Figure 2.9.

Figure 2.9: A standard embedding of $\lvert\Theta\rvert$, along with the
vertical bars from Definition 2.17 that $\theta(\overline{e}_{0})$ is not
allowed to intersect.

######  Definition 2.18.

Let $\theta,e_{0}$ be as in Definition 2.17. As with framed theta-curves, the
direction of the interior of $\theta(\overline{e}_{0})$ perpendicular to
$e_{0}$ determines a framing integer in $\pi_{1}(SO(2))\cong\mathbb{Z}$. To
define the coframing, parametrize $e_{0}$ by $t\in[0,1]$ such that
$e_{0}(0)\in a_{0}$ and $e_{0}(1)\in a_{1}$. Note that $e_{0}$ is a smooth arc
by assumption on $\theta:\lvert\Theta\rvert\hookrightarrow S^{3}$. For
$t\in(0,1)$ and $i\in\\{0,1\\}$, consider the vectors in $S^{3}$ from
$e_{0}(i)$ to $e_{0}(t)$. Project these vectors onto $P$ and normalize them.
This is possible by the fourth assumption of standard-ness: note that without
this assumption some of these projections onto $P$ could yield the zero
vector. This results in paths $p_{i}$ in $SO(2)$, which are loops by the first
and third assumption of standard-ness. Let $n_{i}$ denote the associated
elements of $\pi_{1}(SO(2))\cong\mathbb{Z}$. Then we define the coframing of
$\theta$ to be given by

$\text{cofr}(\theta)=n_{0}-n_{1}.$ (2.4)

It is easy to see that any biframed theta-curve $\theta$ for which
$\theta(\overline{e}_{-}\cup\overline{e}_{+})$ is the unframed unknot can be
put into standard form: moves of a theta-curve away from the attaching points
extend readily to moves of biframed theta-curves. Moves at the attaching
points can be extended by replacing swivels around the end-points by twists
that drag $\overline{e}_{0}$ along, as in the right-hand side of Figure 2.11.
After such an extension, $\theta$ can be put into standard form by applying
moves such as that depicted in the right-hand side of Figure 2.11 to put
$\theta(\overline{e}_{-}\cup\overline{e}_{+})$ into standard unframed form,
followed by deformations in regular neighbourhoods of $a_{0},a_{1}$ and
deformations moving $\theta(\overline{e}_{0})$ away from the bars of Figure
2.9.

######  Remark 2.19.

As before it is clear that $\text{fr}(\theta)$ is independent of how $\theta$
is put into standard form, but this is not immediate for
$\text{cofr}(\theta)$: the ambiguity arises because we can undo a twist in
$\theta(\overline{e}_{-}\cup\overline{e}_{+})$ either at $a_{0}$ or at
$a_{1}$, dragging $\overline{e}_{0}$ around one end of
$\theta(\overline{e}_{-}\cup\overline{e}_{+})$ or the other. This is
illustrated in Figure 2.10.

Figure 2.10: Two ways to bring a $(0,1)$-biframed theta-curve into standard
form.

As can be read off immediately from Figure 2.10, the difference between
undoing a twist in $\theta(\overline{e}_{-}\cup\overline{e}_{+})$ at one end
or the other is a difference $n_{i}\to n_{i}-1$ for $i\in\\{0,1\\}$. The minus
sign in Equation (2.4) cancels this difference, so that the coframing is
independent of the chosen standard form. Analogous reasoning holds for
oppositely directed twist in $\theta(\overline{e}_{-}\cup\overline{e}_{+})$.
Thus Definition 2.18 gives a well-defined coframing for any biframed theta-
curve $\theta$ such that $\theta(\overline{e}_{-}\cup\overline{e}_{+})$ is the
unframed unknot.

Definition 2.18 and Remark 2.19 allow us to define the framing and coframing
of any biframed theta-curve $\theta$ for which
$\theta(\overline{e}_{-}\cup\overline{e}_{+})$ is the unframed unknot. Using
this we can define the objects that are of interest to us, namely ‘simple’
biframed theta-curves:

######  Definition 2.20.

A biframed theta-curve $\theta$ is simple if the annulus
$\theta(\overline{e}_{-}\cup\overline{e}_{+})$ is the unframed unknot and
$\theta$ has integer framing and coframing numbers.

In particular the example shown in Figure 2.8 is simple.

######  Remark 2.21.

For both standard and simple biframed theta-curves, note that we have
restricted to embeddings of $\lvert\Theta\rvert$ such that
$\overline{e}_{-}\cup\overline{e}_{+}$ has framing $0$. One may think that the
complexity added by making $\overline{e}_{-}\cup\overline{e}_{+}$ a ribbon is
negated by this restriction. To see that this is not the case, consider Figure
2.11.

Figure 2.11: Two standard biframed theta-curves differing only in coframing.

Figure 2.11 depicts two simple biframed theta-curves. Indeed, for the second
theta-curve $\overline{e}_{-}\cup\overline{e}_{+}$ can clearly be deformed
into the standard unframed form (as depicted). However, the deformation that
accomplishes this drags $\overline{e}_{0}$ around
$\overline{e}_{-}\cup\overline{e}_{+}$, as pictured, resulting in a biframed
theta-curve that differs from the first. The coframing is defined to record
exactly such differences.

The upshot of the biframing is the following lemma:

######  Lemma 2.22.

There is a bijection:

$\displaystyle\\{\text{Simple biframed theta-curves}\\}$
$\displaystyle\leftrightarrow\\{\text{Simple theta-
curves}\\}\times\mathbb{Z}^{2}$ (2.5) $\displaystyle\theta$
$\displaystyle\mapsto\left(\theta^{\circ},\text{fr}(\theta),\text{cofr}(\theta)\right).$

Here $\theta^{\circ}$ is the theta-curve given by any one component of the
$1$-dimensional boundary of $\theta(\lvert\Theta\rvert)$.

###### Proof.

This proof is analogous to that of Lemma 2.13. In short: surjectivity is clear
and for injectivity an equivalence $\theta_{1}^{\circ}\cong\theta_{2}^{\circ}$
extends to $\theta_{1}\cong\theta_{2}$ up to framing twists and coframing
loops that must cancel if $\text{fr}(\theta_{1})=\text{fr}(\theta_{2})$ and
$\text{cofr}(\theta_{1})=\text{cofr}(\theta_{2})$ (e.g. via Figure 2.10). ∎

####  2.2.4 Biframed Knotoids

Next we define biframed knotoids. As the terminology suggests, this will turn
out to be the class of diagrams whose geometric realization is
$\lvert\Theta\rvert$. In light of Remark 2.15 and the proof of Theorem 2.5, it
is no surprise that these are the knotoid diagrams obtained from projecting
the smooth arc $e_{0}$ of a standard biframed theta-curve onto its plane $P$.

######  Definition 2.23.

Pick two points $v_{0},v_{1}\in S^{2}$. Denote the directed straight arc in
$\mathbb{R}^{2}\cup\\{\infty\\}\cong S^{2}$ from $v_{0}$ to $v_{1}$ by $L$. A
biframed knotoid diagram is a knotoid diagram $K$ in $S^{2}$ with leg and head
equal to $v_{0}$ and $v_{1}$ respectively, such that the tangents of $K$
coincide with $L$ in neighbourhoods of $v_{0},v_{1}$. Two biframed knotoid
diagrams are said to be equivalent if they can be related by applications of
$R1^{\prime},R2,R3,R4$, ambient isotopies of $S^{2}$ away from $v_{0},v_{1}$,
and the coframing identities depicted in Figure 2.12 which encode the move in
Figure 2.10.

Figure 2.12: The coframing identities. Here $K$ denotes the rest of the
knotoid diagram, which is fixed under the identities.

Fixing $v_{0},v_{1}$ uniformly for all biframed knotoid diagrams prevents the
case of biframed knotoid diagrams that are in-equivalent only because their
end-points disagree and cannot be moved. An example of a biframed knotoid
diagram is given in Figure 2.13.

Figure 2.13: Example of a biframed knotoid diagram.

To give meaning to the above terminology, we define the framing and coframing
of a biframed knotoid diagram. After Definitions 2.8 and 2.18, there are no
surprises here.

######  Definition 2.24.

Let $K$ be a biframed knotoid diagram. We define its framing to be its writhe.
To define its coframing, let $n_{i}$ be the winding number of $K$ around
$v_{i}$ for $i\in\\{0,1\\}$. By assumption on the tangents of $K$,
$n_{i}\in\mathbb{Z}$. In analogy with Equation (2.4) we define the coframing
of $K$ by

$\text{cofr}(K)=n_{0}-n_{1}.$

To see that the coframing is well-defined we must verify that it is invariant
under the moves on biframed knotoid diagrams. This is the content of the
following lemma.

######  Lemma 2.25.

The coframing is a biframed knotoid diagram invariant.

###### Proof.

The only moves for which invariance is not immediate are the non-local moves,
i.e. $R4$ and the coframing identities. The coframing is defined to be
invariant under the coframing identities, namely these identities respectively
change both $n_{0}$ and $n_{1}$ by $\pm 1$, and so these changes cancel in the
definition of $\text{cofr}(K)$. Next say that $R4$ is applied to an arc that
traverses $\alpha_{i}$ radians around $v_{i}$. Then the new arc traverses
$-(2\pi-\alpha_{i})$ radians around $v_{i}$. Noting that

$-(2\pi-\alpha_{0})+(2\pi-\alpha_{1})=\alpha_{0}-\alpha_{1},$

we see that these differences again cancel in the definition of
$\text{cofr}(K)$. ∎

######  Lemma 2.26.

There is a bijection

$\displaystyle\\{\text{Biframed knotoids}\\}$
$\displaystyle\leftrightarrow\\{\text{Knotoids}\\}\times\mathbb{Z}^{2}$ (2.6)
$\displaystyle K$
$\displaystyle\mapsto\left(K,\text{writhe}(K),\text{cofr}(K)\right).$

###### Proof.

Clearly $n_{0}$, $n_{1}$, and hence the coframing of a biframed knotoid, can
be adjusted at will by adding loops around $v_{0}$ as needed. Combined with
bijection (2.1), this gives surjectivity of assignment (2.6). Injectivity
follows from bijection (2.1), and reasoning analogous to the proof of
bijection (2.1) for the coframing: in short, an equivalence $K\simeq
K^{\prime}$ as knotoids can be extended to an equivalence of biframed
knotoids, up to loops around $v_{0},v_{1}$ which must cancel via Figures 2.12
and 2.14 if we assume $\text{cofr}(K)=\text{cofr}(K^{\prime})$. Thus
assignment (2.6) is a bijection. ∎

Figure 2.14: Cancellation of opposite coframing loops.

Combined with bijection (2.5) and Theorem 2.5 this results in a geometric
realization result for biframed knotoids analogous to bijection (2.3); namely
a bijection

$\\{\text{Biframed knotoids}\\}\leftrightarrow\\{\text{Biframed simple theta-
curves}\\},$ (2.7)

casting $\lvert\Theta\rvert$ as the geometric realization of biframed
knotoids.

As an extension to Remark 2.15, bijection (2.7) also has a topological
realization analogous to the ‘projection’ $\theta\mapsto e_{0}$ from Figure
2.3. The respective definitions of framing for diagrams and theta-curves
translate to each other under this projection via the blackboard framing as
before, and the respective definitions of coframing are set up so that they
translate to each other immediately under this projection.

####  2.2.5 Half-integer Coframing

Our original motivation for introducing biframed theta-curves was finding a
way of adding a ‘half-twist’ to a framed knotoid in such a way that can be
seen easily in the corresponding knotoid diagrams. Here we discuss how
biframed knotoids provide this feature by extending the discussion to non-
integer coframing.

Biframed knotoids with half-integer coframing are obtained simply by weakening
the condition of simplicity of a biframed theta-curves by removing the
requirement that the coframing be integer. This adds the possibility of theta-
curves with $\theta(\overline{e}_{0})$ attached to
$\theta(\overline{e}_{-}\cup\overline{e}_{+})$ at the back. See Figure 2.15.

Figure 2.15: A standard biframed theta-curve with half-integer coframing.

Such an attachment at the back can occur at either end of
$\theta(\overline{e}_{0})$, and we can translate between these two cases via
the isotopies shown in Figure 2.16. These same isotopies show that a biframed
theta-curve with two half-integer coframing attachments is nothing but an
integer-coframing theta-curve, as the terminology ‘half-integer’ suggests.

Figure 2.16: Translating a half-integer coframing attachment from $a_{1}$ to
$a_{0}$. (A re-coloring occurs in the last equivalence.)

The coframing of a half-integer coframing theta-curve is defined analogously
as for the integer-coframing case, i.e. by putting the theta-curve in standard
form and subtracting the turning numbers of $\theta(\overline{e}_{0})$ around
its attachments to $\theta(\overline{e}_{-}\cup\overline{e}_{+})$.

Half-integer coframing manifests in biframed knotoid diagrams as the
possibility of the tangent vectors at the end-points to be opposite to $L$.
See Figure 2.17. (Again, the coframing in this case is defined as a difference
of turning numbers, exactly as for integer-coframing knotoid diagrams.)

Figure 2.17: A $(0,-1/2)$-biframed trivial knotoid diagram.

To properly define half-integer coframing knotoids, then, we need to add
coframing identities encoding the moves in Figure 2.16 and similar such moves.

Figure 2.18: Additional coframing identities for half-integer coframing
knotoids, for coframing $+\frac{1}{2}$, $0$, and $+1$ respectively. Here $K$
denotes the rest of a knotoid diagram as before.

There are a total of 6 coframing identities that need to be added; the three
identities involving a positive half-integer coframing attachment at the leg
are depicted in Figure 2.18.

##  3 Quantum Knotoid Invariants

In this section we begin by recalling the construction of quantum knot
invariants, specifically Reshetikhin-Turaev invariants, and then give several
generalizations of such invariants to the setting of knotoids using framed and
biframed knotoid diagrams. One such generalization relies on a slight
modification to the setup of quantum invariants, moving to the setting of
‘braided groups’. Hence we first recall this setup very briefly.

###  3.1 Reshetikhin-Turaev Invariants

We follow standard notation from [13], where $\mathcal{C}$ is a category,
$\otimes$ provides a monoidal structure on $\mathcal{C}$ with unit $k$,
$(\Psi,\nu)$ further provide a ribbon braided structure on $\mathcal{C}$, and
$(\cdot)^{*}$ is a dualization operation on $\mathcal{C}$ that turns it into a
rigid category via $(\text{ev},\text{coev})$. Recall that quantum knot
invariants are constructed by fixing a rigid ribbon category
$(\mathcal{C},\otimes,k,\Psi,(\cdot)^{*},\text{ev},\text{coev},\nu)$, picking
an object $V$ of $\mathcal{C}$, and mapping a knot diagram to an element of
$\operatorname{Hom}_{\mathcal{C}}(k,k)$ via the assignment specified on
elementary tangles by Figure 3.1. This assignment, call it $\varphi$, is
extended to arbitrary tangle diagrams via the rules $\varphi(T_{1}\otimes
T_{2})=\varphi(T_{1})\otimes\varphi(T_{2})$ and $\varphi(T_{1}\circ
T_{2})=\varphi(T_{1})\circ\varphi(T_{2})$, where the tensor product and
composition of tangles denote their horizontal and vertical juxtapositions
respectively. For a more detailed discussion see [10].

Figure 3.1: The assignment $\varphi$ on elementary tangles.

Here $\overline{\text{ev}}_{V}$ and $\overline{\text{coev}}_{V}$ are defined
using the ribbon element $\nu_{V}^{-1}$, as is detailed for
$\overline{\text{coev}}_{V}$. Our convention is that knot diagrams are read
top to bottom, with downward strands represented by $V$ and upward strands by
$V^{*}$. More generally for crossings that involve $V^{*}$, $\Psi$ is assigned
to positively oriented crossings and $\Psi^{-1}$ to negative ones.

The assignment $\varphi$ is well-defined on framed tangles, and therefore
provides an invariant of oriented framed knots for any choice
$(\mathcal{C},V)$ when applied to a knot diagram viewed as a tangle with no
in- or out-going strands. Such an invariant is computed by selecting a ‘Morse
decomposition’ of the knot; dividing it into a composition of tangles, each of
which is a finite tensor product of elementary tangles.

For our purposes here, we will wish to extend $\varphi$ to the setting of
tangle diagrams decorated with splittings and mergings of strands. To this
end, consider a Hopf algebra $(H,\cdot,\Delta,\eta,\epsilon,S)$, following
standard notation from [12]. Then in analogy with $\varphi$ we can represent
morphisms $H^{\otimes n}\to H^{\otimes m}$ as tangles decorated with
splittings, mergings, and end-points as in Figure 3.2.

Figure 3.2: Diagrammatic depiction of structure morphisms of $H$.

To illustrate this, the resulting diagrammatic identities representing the
axioms of a Hopf algebra are depicted in Figure 3.3.

Figure 3.3: The axioms of a Hopf algebra in diagrammatic notation.

Notice that the second-to-last axiom in Figure 3.3 contains a crossing. For
Hopf algebras this represents the vector space isomorphism $\tau:V\otimes
W\cong W\otimes V$, i.e. the trivial braiding on the category
$\texttt{Vect}_{k}$. The diagrammatic notation developed so far now suggests
we define the following:

######  Definition 3.1.

[11, 12] Let $\mathcal{C}$ be a ribbon category. A braided group (or ‘Hopf
algebra object of a ribbon category’) is an object $B$ of $\mathcal{C}$
endowed with morphisms $\cdot:B\otimes B\to B$, $\Delta:B\to B\otimes B$,
$\eta:k\to B$, $\epsilon:B\to k$, $S:B\to B$. These morphisms must be such
that the axioms depicted in Figure 3.3 are satisfied when considered as
equations of morphisms via Figures 3.1 and 3.2. This means the crossing in
Figure 3.3 is now taken to represent $\Psi$ instead of $\tau$, in accordance
with Figure 3.1.

If $B$ is a braided group in $\mathcal{C}$, then the inverse of Figure 3.2
provides an extension of $\varphi$ to tangle diagrams decorated with
splittings, mergings, and end-points. This extension is well-defined on
decorated framed tangles because $\varphi$ is, and because we can pull the
decorations under and over crossings by naturality of $\Psi,\Psi^{-1}$.

Finally recall that to produce explicit Reshetikhin-Turaev invariants one
usually picks a ribbon Hopf algebra $H$ and picks
$\mathcal{C}={}_{H}\mathcal{M}_{F}$, the category of finite-dimensional
$H$-modules. By the representation theory of Hopf algebras this is a rigid
ribbon category [12, Ch. 9]. A closely related choice is
$\mathcal{C}={}_{H}^{H}\mathcal{M}_{F}$, the category of finite-dimensional
crossed $H$-modules, which is equivalent to ${}_{D(H)}\mathcal{M}_{F}$ for $H$
finite-dimensional where $D(H)$ is the Drinfeld double of $H$ [13, Ch. 10].

###  3.2 Group-Based Invariants

In this subsection we generalize quantum invariants resulting from seeing a
finite-dimensional Hopf algebra $H$ as a crossed module in
${}_{H}^{H}\mathcal{M}_{F}={}_{D(H)}\mathcal{M}_{F}$. We restrict to $H=kG$
for $k$ a field and $G$ a finite group because it will give us an additional
knotoid invariant for free, but the contents of this subsection generalize
readily to arbitrary (finite-dimensional) $H$ as in [16].

For $H=kG$, we first describe an alternative way of computing the framed knot
invariant associated to $H\in{}_{H}^{H}\mathcal{M}_{F}$. Call this invariant
$V_{G}$. Note that $D(kG)=(kG)^{*}\otimes kG\cong k(G)\otimes kG$, where
$k(G)$ is the group of functions $G\to k$, and multiplication in $D(kG)$ is
given by

$(\delta_{a}\otimes h)(\delta_{b}\otimes
g)(x)=\delta_{a}\delta_{hbh^{-1}}\otimes hg$ (3.1)

where $g,h,a,b\in G$, and $\delta_{a}$ is the Kronecker delta function taking
value 1 on $a$. Moreover $D(kG)$ has quasitriangular structure $\mathcal{R}$
given by

$\mathcal{R}=\sum_{g\in G}(\delta_{g}\otimes e)\otimes(1\otimes g)\qquad\text{
and }\qquad\mathcal{R}^{-1}=\sum_{g\in G}(\delta_{g^{-1}}\otimes
e)\otimes(1\otimes g).$

Here $e\in G$ is the group unit. For more details see [12, Ch. 7]. Note that
each term of $\mathcal{R}\in D(H)\otimes D(H)$ consists of a ‘left part’
$(\delta_{g}\otimes e)$ and a ‘right part’ $(1\otimes g)$; similarly for
$\mathcal{R}^{-1}$.

Now let $K$ be a framed knot. Present $K$ as an oriented long knot diagram.
Then $V_{G}(K)$ is an element of
$\operatorname{Hom}(kG,kG)\cong(kG)^{*}\otimes kG=D(kG)$. To compute this
element we place at every crossing of $K$ a copy of $\mathcal{R}$ if the
crossing is positive, or a copy of $\mathcal{R}^{-1}$ if it is negative. We
then trace through the long knot diagram of $K$ following its orientation.
Beginning with the unit $1\otimes e\in D(kG)$, every time we encounter a
crossing we multiply (from the left) with the left part of $\mathcal{R}$ or
$\mathcal{R}^{-1}$ if we are going under the crossing, or with the right part
if we are going over. After the final crossing we sum over all indices of all
crossings, and the result is equal to $V_{G}(K)$ [16]. To obtain the knot
invariant associated to this long knot invariant one takes the trace of
$V_{G}(K)\in\operatorname{Hom}(kG,kG)$. The result is the cardinality of
$\operatorname{Hom}(\pi_{1}(S^{3}-K),G)$ [13, Ch. 13], for reasons we shall
soon discuss.

As we shall see in the next subsection, the general Reshetikhin-Turaev
invariant construction is tricky to generalize to knotoids. However, the above
algorithm for computing $V_{G}$ generalizes effortlessly to give a framed
knotoid invariant:

######  Definition 3.2.

Let $K$ be a diagram for a framed knotoid. Recall that all knotoids are
implicitly oriented from leg to head. Define $V_{G}(K)\in D(H)$ by the same
process as above, tracing through $K$ from leg to head.

######  Proposition 3.3.

The element $V_{G}(K)$ is a framed knotoid invariant.

###### Proof.

Invariance under ambient isotopy and the spherical move $R4$ is immediate
since these moves do not alter the crossing data of $K$. Thus it suffices to
check invariance of $V_{G}(K)$ under $R1^{\prime},R2,R3$. This follows from
$V_{G}$ being an invariant when defined on framed (long) knots. For
completeness, we also prove by hand that $V_{G}$ is a framed knotoid invariant
below:

Since the oriented versions of $R1^{\prime}$ do not change the signs of
crossings, invariance of $V_{G}(K)$ under $R1^{\prime}$ follows from

$(\delta_{g}\otimes e)(1\otimes g)=\delta_{g}\otimes g=(1\otimes
g)(\delta_{g}\otimes e)\hskip 5.0pt\text{ and }\hskip
5.0pt(\delta_{g^{-1}}\otimes e)(1\otimes g)=\delta_{g^{-1}}\otimes g=(1\otimes
g)(\delta_{g^{-1}}\otimes e).$

The move $R2$ has four oriented versions. The move $R3$ has eight. For brevity
we consider only one of each; see Figure 3.4. The proofs for the other
orientations are analogous.

Figure 3.4: Examples of oriented $R2$ and $R3$ moves.

On the left-hand side of the $R2$ move in Figure 3.4 there are two crossings.
For each orientation of $R2$, one is positive and the other negative. To find
invariance we must show that the contributions from these crossings cancel.
Indeed, the bottom arc gives a contribution

$(\delta_{g^{-1}}\otimes e)(\delta_{h}\otimes
e)=\begin{cases}\delta_{h}\otimes e&\quad\text{ if }g^{-1}=h,\\\ 0&\quad\text{
otherwise.}\end{cases}$

The top arc gives a contribution $(1\otimes g)(1\otimes h)$ which is thus
$(1\otimes e)$ for nonzero contributions. Noting that
$\sum_{h}(\delta_{h}\otimes e)=1\otimes e$ we see that the contribution of the
bottom arc is also trivial, as required.

To verify invariance under the $R3$ move in Figure 3.4, we calculate the
contributions of both sides and show they are equal. When running through $K$,
suppose we pass through the vertical line first, then through the line going
diagonally up, and finally through the strand going diagonally down (other
orders are proven analogously). Then we compute $V_{G}(K)$ on the left-hand
side to be

$\displaystyle V_{G}(K)$ $\displaystyle=\sum_{g,h,k,\dots}\dots(1\otimes
g)(1\otimes h)\dots(\delta_{g^{-1}}\otimes e)(1\otimes
k)\dots(\delta_{h^{-1}}\otimes e)(\delta_{k^{-1}}\otimes e)$
$\displaystyle=\sum_{g,k,\dots}\dots(1\otimes g)(1\otimes
k)\dots(\delta_{g^{-1}}\otimes e)(1\otimes k)\dots(\delta_{k^{-1}}\otimes e).$

Here the $(\dots)$ are generic placeholder for contributions due to the rest
of $K$. Now we drag the term $(\delta_{k^{-1}}\otimes e)$ towards the middle
contribution. Every time an element of the form $(1\otimes x)$ is encountered,
the argument of $\delta_{k^{-1}}$ is conjugated by $x^{-1}$. Thus we find

$\displaystyle V_{G}(K)$ $\displaystyle=\sum_{g,k,\dots}\dots(1\otimes
g)(1\otimes k)\dots(\delta_{g^{-1}}\otimes e)(1\otimes
k)(\delta_{wk^{-1}w^{-1}}\otimes e)\dots$
$\displaystyle=\sum_{g,k,\dots}\dots(1\otimes g)(1\otimes
k)\dots(\delta_{g^{-1}}\otimes e)(\delta_{kwk^{-1}w^{-1}k^{-1}}\otimes
e)(1\otimes k)\dots,$

for some word $w$ on $G$. This implies that $g^{-1}=kwk^{-1}w^{-1}k^{-1}$.
Thus we conclude

$V_{G}(K)=\sum_{k,\dots}\dots(1\otimes
kwkw^{-1})\dots(\delta_{kwk^{-1}w^{-1}k^{-1}}\otimes k)\dots$

Following the same procedure on the right-hand side (with the labels $g,h,k$
attached to the same crossings as before) we find

$\displaystyle V_{G}(K)$ $\displaystyle=\sum_{g,k,\dots}\dots(1\otimes
k)(1\otimes g)\dots(1\otimes k)(\delta_{g^{-1}}\otimes
e)\dots(\delta_{k^{-1}}\otimes e)$
$\displaystyle=\sum_{g,k,\dots}\dots(1\otimes k)(1\otimes g)\dots(1\otimes
k)(\delta_{g^{-1}}\otimes e)(\delta_{wk^{-1}w^{-1}}\otimes e)\dots$
$\displaystyle=\sum_{k,\dots}\dots(1\otimes kwkw^{-1})\dots(1\otimes
k)(\delta_{wk^{-1}w^{-1}}\otimes e)\dots$
$\displaystyle=\sum_{k,\dots}\dots(1\otimes
kwkw^{-1})\dots(\delta_{kwk^{-1}w^{-1}k^{-1}}\otimes k)\dots.$

(Note that $w$ here is the same word as in the left-hand side computation!) We
thus conclude that $V_{G}$ is indeed invariant under $R3$, completing the
proof. ∎

######  Example 3.4.

Let $K$ be the framed knotoid depicted in Figure 3.5.

Figure 3.5: Construction of $V_{G}$ for an example framed knotoid.

The signs of the crossings of $K$ are indicated in Figure 3.5, as are the
nontrivial parts of $\mathcal{R},\mathcal{R}^{-1}$ that occur in the
construction of $V_{G}$. Using the definition of $V_{G}$ and Equation (3.1) we
compute:

$\displaystyle V_{G}(K)$ $\displaystyle=\sum_{g,h,k\in G}(\delta_{g}\otimes
e)(1\otimes h)(\delta_{h^{-1}}\otimes e)(1\otimes k)(1\otimes
g)(\delta_{k}\otimes e)$ $\displaystyle=\sum_{g,h,k\in
G}\delta_{g}\delta_{h^{-1}}\delta_{hkgkg^{-1}k^{-1}h^{-1}}\otimes hkg$
$\displaystyle=\sum_{g,k\in G}\delta_{g}\delta_{g^{-1}kgkg^{-1}k^{-1}g}\otimes
g^{-1}kg$ $\displaystyle=\sum_{g,k\in G}\delta_{gkg}\delta_{kgk}\otimes
g^{-1}kg.$

Here the last two equalities use that $\delta_{a}\delta_{b}\neq 0\iff a=b$.
This also means that a choice of $(g,k)$ only gives a nonzero contribution to
$V_{G}(K)$ if $gkg=kgk$. In other words, we only obtain a contribution to
$V_{G}(K)$ if $B_{3}\to G:(x,y)\mapsto(g,k)$ constitutes a group homomorphism,
where

$B_{3}=\langle x,y\,|\,xyx=yxy\rangle$

is the braid group on $3$ strands.

At the beginning of this subsection we stated that taking $H=kG$ gives us an
extra knotoid invariant for free. Indeed, this follows from the relation
between $V_{G}$ and the Wirtinger presentation of the knot group
$\pi_{1}(S^{3}-K)$. See [14, Ch. 3] for more on the Wirtinger presentation.
Namely, if $K$ is a knot presented as a long knot diagram then after some
simplification $V_{G}(K)$ is of the form

$V_{G}(K)=\sum_{g,h,\dots}\delta_{\dots}\delta_{\dots}\dots\delta_{\dots}\otimes\dots.$

Here the subscripts of the $\delta$’s are words in $(g,h,\dots)$. We only get
a non-zero contribution to $V_{G}$ for $g,h,\dots$ such that all these words
coincide. The multiplication in $D(H)$ is set up precisely so that the
relations corresponding to coincidence of these words are exactly the
relations of the Wirtinger presentation. Indeed: if we encounter a term in
$V_{G}$ such as

$(\delta_{k}\otimes e)(1\otimes g)(\delta_{h}\otimes e),$

then to arrive at the reduced form for $V_{G}$ we must pull the $\delta_{h}$
term to the left. The multiplication in $D(H)$ prescribes that this term
equals

$(\delta_{k}\otimes e)(\delta_{ghg^{-1}}\otimes
g)=\delta_{k}\delta_{ghg^{-1}}\otimes g.$

This conjugation by $g$ reflects exactly the action of going under the over-
lying $g$-strand and back again in order to relate $h$ to $k$, which is
precisely the setup of the Wirtinger presentation.

In conclusion, the invariant $V_{G}$ gives us an alternative way of computing
the Wirtinger presentation. This implies that nonzero contributions to
$V_{G}(K)$ correspond to homomorphisms $\pi_{1}(S^{3}-K)\to G$. Both of these
results are well-known [1]. The novelty is that we have extended $V_{G}$ to
framed knotoids, and so this interpretation of $V_{G}$ allows us to associate
a group to knotoids.

######  Definition 3.5.

Let $K$ be a knotoid diagram with $n$ crossings. Suppose $V_{G}(K)$ is of the
form

$V_{G}(K)=\sum_{g_{1},g_{2},\dots,g_{n}}\delta_{w_{1}}\delta_{w_{2}}\dots\delta_{w_{n}}\otimes
w_{n+1}$

where each $w_{i}$ is a word in $g_{1},\dots g_{n}$. We define the knotoid
group $W(K)$ of $K$ by the presentation:

$W(K)=\langle g_{1},g_{2},\dots g_{n}|w_{1}=w_{2}=\dots=w_{n}\rangle.$

######  Remark 3.6.

One might expect the knotoid group $W(K)$ to be isomorphic to the fundamental
group of $S^{3}$ with an arc that projects to give $K$ removed, in which case
it is isomorphic to $S^{3}$ with an arc removed and hence trivial. Instead
$W(K)$ is a group defined by the combinatorial information of $K$ and hence
not necessarily related to any fundamental group. We shall see below that it
is nontrivial in general. A knotoid group was also defined by Turaev directly
via the Wirtinger presentation [15]. Turaev’s knotoid group, which I shall
denote here by $W^{\prime}(K)$, has one generator more than $W(K)$, and in
[15] it is shown that $W^{\prime}(K)\cong\pi_{1}(S^{3}-K_{-})$ where $K_{-}$
is the knot formed from a knotoid $K$ by connecting the endpoints via a strand
that runs under all of $K$. We shall see in Example 3.8 below that this last
fact does not hold for $W(K)$, and hence $W(K)\ncong W^{\prime}(K)$ in
general.

Since $V_{G}$ is a framed knotoid invariant, invariance of $W(K)$ under
$R1^{\prime},R2,R3$ is immediate. Note however that Definition 3.5 makes no
reference to framing. This is justified by the following lemma:

######  Lemma 3.7.

The knotoid group $W(K)$ is $R1$-invariant.

###### Proof.

Let  and  denote two knotoid diagrams for $K$ that are equal outside of the
depicted discs. Let us write

$\displaystyle
V_{G}\big{(}\raisebox{-6.82881pt}{\includegraphics[width=19.91684pt]{WKleft.pdf}}\big{)}$
$\displaystyle=\sum_{g,g_{1},g_{2},\dots}\dots(1\otimes
g)(\delta_{g^{-1}}\otimes e)\dots$
$\displaystyle=\sum_{\dots,g,g_{1},\dots}\left(\delta_{\dots}\dots\delta_{\dots}\delta_{\Omega
g^{-1}\Omega^{-1}}\delta_{\Omega
gw_{1}g_{1}w_{1}^{-1}g^{-1}\Omega^{-1}}\delta_{\Omega
gw_{2}g_{2}w_{2}^{-1}g^{-1}\Omega^{-1}}\dots\right)\otimes w^{\prime}.$

Here $w^{\prime},w_{1},w_{2},\dots$ are words in $g_{1},g_{2},\dots$, and
$\Omega$ is the word by which the term $\delta_{g^{-1}}$ gets conjugated in
the reduction of $V_{G}$. We want to show that the generator $g$ can be
removed from the defining presentation of $W(K)$. Omitting the generators from
the presentations, we compute

$\displaystyle
W\big{(}\raisebox{-6.82881pt}{\includegraphics[width=19.91684pt]{WKleft.pdf}}\big{)}$
$\displaystyle=\langle\dots=\Omega g^{-1}\Omega^{-1}=\Omega
gw_{1}g_{1}w_{1}^{-1}g^{-1}\Omega^{-1}=\Omega
gw_{2}g_{2}w_{2}^{-1}g^{-1}\Omega^{-1}=\dots\rangle$
$\displaystyle=\langle\dots=\Omega g^{-1}\Omega^{-1},$
$\displaystyle\qquad\qquad
g^{-1}=gw_{1}g_{1}w_{1}^{-1}g^{-1}=gw_{2}g_{2}w_{2}^{-1}g^{-1}=\dots\rangle$
$\displaystyle=\langle\dots=\Omega g^{-1}\Omega^{-1},$
$\displaystyle\qquad\qquad
g^{-1}=w_{1}g_{1}w_{1}^{-1}=w_{2}g_{2}w_{2}^{-1}=\dots\rangle$
$\displaystyle=\langle\dots=\Omega w_{1}g_{1}w_{1}^{-1}\Omega^{-1},$
$\displaystyle\qquad\qquad
w_{1}g_{1}w_{1}^{-1}=w_{2}g_{2}w_{2}^{-1}=\dots\rangle$
$\displaystyle=\langle\dots=\Omega w_{1}g_{1}w_{1}^{-1}\Omega^{-1}=\Omega
w_{2}g_{2}w_{2}^{-1}\Omega^{-1}=\dots\rangle=W\big{(}\raisebox{-6.82881pt}{\includegraphics[width=19.91684pt]{WKright.pdf}}\big{)},$

as required. The proof for the $R1$-relation of the other sign is analogous. ∎

######  Example 3.8.

For the knotoid diagram $K$ from Example 3.4, we saw $W(K)=B_{3}$. Note that
$K_{-}$ is equivalent to the unknot, and hence
$W^{\prime}(K)\cong\pi_{1}(S^{3}-K_{-})\cong\mathbb{Z}$ where $W^{\prime}(K)$
is the knotoid group defined in [15], so that $W^{\prime}(K)\ncong W(K)$ in
this case.

######  Example 3.9.

As a more substantial example of computing $V_{G}$, we shall show that
$V_{G}(K)$ can contain more information than the cardinality of
$\text{Hom}(W(K),G)$. Indeed: let $K_{1}$ and $K_{2}$ be the oriented knotoids
depicted in figure 3.6.

Figure 3.6: The knotoids $K_{1}$ and $K_{2}$, corresponding respectively to
orientations of the knotoids $5_{5}$ and $5_{24}$ in the knotoid table from
[6].

We compute that

$\displaystyle V_{G}(K_{1})$ $\displaystyle=\sum_{g,h,i,j,k\in
G}(\delta_{g^{-1}}\otimes e)(\delta_{h}\otimes e)(1\otimes
i)(\delta_{j}\otimes e)(1\otimes g)$ $\displaystyle\hskip
86.72267pt\cdot(1\otimes j)(1\otimes k)(\delta_{i}\otimes e)(1\otimes
h)(\delta_{k^{-1}}\otimes e)$ $\displaystyle=\sum_{g,h,i,j,k\in
G}\delta_{g^{-1}}\delta_{h}\delta_{iji^{-1}}\delta_{igjki(igjk)^{-1}}\delta_{igjkhk^{-1}(igjkh)^{-1}}\otimes
igjkh.$

and

$\displaystyle V_{G}(K_{2})$ $\displaystyle=\sum_{g,h,i,j,k\in
G}(\delta_{g}\otimes e)(1\otimes h)(1\otimes i)(\delta_{j^{-1}}\otimes
e)(\delta_{k^{-1}}\otimes e)$ $\displaystyle\hskip 86.72267pt\cdot(1\otimes
g)(1\otimes k)(\delta_{h^{-1}}\otimes e)(1\otimes j)(\delta_{i^{-1}}\otimes
e)$ $\displaystyle=\sum_{g,h,i,j,k\in
G}\delta_{g}\delta_{hij^{-1}(hi)^{-1}}\delta_{hik^{-1}(hi)^{-1}}\delta_{higkh^{-1}(higk)^{-1}}\delta_{higkji^{-1}(higkj)^{-1}}\otimes
higkj.$

Taking $G=S_{3}$, we find that both $V_{S_{3}}(K_{1})$ and $V_{S_{3}}(K_{2})$
have 12 nonzero contributions. More specifically, we find

$\displaystyle V_{S_{3}}(K_{1})=\delta_{e}\otimes
e+\delta_{(1,3,2)}\otimes(1,3,2)+\delta_{(1,2,3)}\otimes(1,2,3)+\dots,$
$\displaystyle V_{S_{3}}(K_{2})=\delta_{e}\otimes e+\delta_{(1,2,3)}\otimes
e+\delta_{(1,3,2)}\otimes e+\dots,$

where the omitted terms in both expressions each consist of ten terms only
involving the 2-cycles $(1,2),(1,3),(2,3)\in S_{3}$. From these computations
we note that

$\\#\text{Hom}(W(K_{1}),S_{3})=12=\\#\text{Hom}(W(K_{2}),S_{3}),$

meaning that $\\#\text{Hom}(W(\cdot),S_{3})$ cannot distinguish $K_{1}$ and
$K_{2}$. However, the expressions of $V_{S_{3}}(K_{1})$ and $V_{S_{3}}(K_{2})$
given above clearly show that $V_{S_{3}}(K_{1})\neq V_{S_{3}}(K_{2})$, which
implies that $V_{S_{3}}(\cdot)$ can distinguish $K_{1}$ and $K_{2}$. We
therefore conclude that $V_{G}(\cdot)$ can be a strictly stronger invariant
than $\\#\text{Hom}(W(\cdot),S_{3})$, as claimed.

###  3.3 Reshetikhin-Turaev Invariants of Knotoids

There are two main obstructions to extending general Reshetikhin-Turaev
invariants to framed knotoids. The first point is of course that of
determining what to do with the end-points. The second obstacle is that for a
(framed) knotoid diagram, one is able to swivel the tangent vector at the end-
points to go, say, either up or down. Such swiveling affects the number of ’s
and ’s in a Morse decomposition of the diagram. Unfortunately there is no
reason that a Reshetikhin-Turaev-type assignment of morphisms to knotoid
diagrams should be invariant under this action.

Fortunately, in light of Section 2 we are equipped to solve this second
problem. Namely, restricting our scope to producing invariants of biframed
knotoids removes the obstruction entirely. A posteriori this is one of the
main reasons for being interested in biframed knotoids in the first place!

Naturally, the first obstruction admits several solutions. Intuitively one
would think to extend $\varphi$ to knotoid diagrams by assigning to  and
morphisms $k\to V$ and $V\to k$ respectively. However, by naturality of $\Psi$
and $\Psi^{-1}$, $\varphi$ would then be invariant under the forbidden moves
in Figure 1.1 causing the associated invariant to be trivial.

Here we consider two options of how to handle  and .

####  3.3.1 Option 1

The next best thing to representing  and  by morphisms $k\to V$ or $V\to k$,
is to represent them by linear maps that are not morphisms.

######  Definition 3.10.

Suppose $\mathcal{C}={}_{H}\mathcal{M}_{F}$ for some ribbon Hopf algebra111By
Tannaka duality this is equivalent to $\mathcal{C}$ having a fiber functor
[12, Ch. 9]. $H$. Let $V$ be an $n$-dimensional object of $\mathcal{C}$, and
$K$ a biframed knotoid diagram. For simplicity we assume $K$ has integer
coframing. View $K$ as a tangle decorated with two endpoints representing maps
$\eta:k\to V$ and $\epsilon:V\to k$. Extend $\varphi$ to an assignment on $K$,
by choosing a basis $\\{e_{1},\dots,e_{n}\\}$ of $V$ and defining

$\eta:1\mapsto\sum_{i=1}^{n}c_{i}e_{i}\qquad\text{ and
}\qquad\epsilon:\sum_{i=1}^{n}\lambda_{i}e_{i}\mapsto\sum_{i=1}^{n}\lambda_{i}d_{i},$

where $\\{c_{i},d_{i}\\}_{i=1}^{n}$ consists of elements of $k$. The maps
$\eta$ and $\epsilon$ are chosen to be linear maps, but not $H$-module
isomorphisms. The Reshetikhin-Turaev invariant associated to $K$ is then
$\varphi(K)$.

This solution is closely related to the approach taken in [9]. It has the
advantage of being a simple and natural extension of $\varphi$. A distinct
disadvantage is that this construction has no reason a priori to be invariant
under the coframing identities, nor under the spherical move of Remark 2.2.
Thus in general this approach yields invariants of planar ‘triframed’ knotoid
diagrams, meaning that both endpoints have separate coframings which are not
related by a coframing identity.

If we allow the end-points of these triframed knotoids to move around on
$\mathbb{R}^{2}$, keeping the tangent vectors at the leg and head fixed, then
we end up with the ‘Morse knotoids’ from [9]. It turns out that the
construction of quantum invariant given here also yields invariants of Morse
knotoids, and that these Morse knotoids are essentially biframed planar
knotoids in the sense that they can be described by knotoid type, framing
integer, and ‘rotation number’. The rotation number is equivalent to the
coframing as defined for diagrams in Definition 2.24. See [9] for more
details. To keep the discussion below self-contained, we will work with
triframed knotoid diagrams.

######  Example 3.11.

We consider the example of $H=U_{q}(\mathfrak{sl}_{2})$, letting $V$ be the
standard 2-dimensional representation. In this case the structures of
$\Psi,\text{ev},\text{coev}$, etc. are well-known (up to a choice of
normalization); see e.g. [4, Ch. 2.6] or [10, Ch. 9.4]. Let
$\\{e_{1},e_{2}\\}$ be the standard basis on $V$, $\\{e^{1},e^{2}\\}$ the dual
basis on $V^{*}$, and $\\{e_{1}\otimes e_{1},e_{1}\otimes e_{2},e_{2}\otimes
e_{1},e_{2}\otimes e_{2}\\}$ the standard ordered basis of $V\otimes V$ (and
similarly for $V^{*}\otimes V$, $V\otimes V^{*}$). Then the values of
$\varphi$ on knot diagram elements with respect to these bases are given in
Figure 3.7.

Figure 3.7: Values of $\varphi$ on knot diagram elements.

From now on, assume for simplicity that we are working with a (triframed)
knotoid diagram $K$ with integer coframing, drawn so that the tangent vectors
at the end-points of $K$ are vertical.

We extend $\varphi$ to triframed knotoid diagrams by associating undetermined
linear maps to the endpoints; see Figure 3.8.

Figure 3.8: Values of $\varphi$ on remaining knotoid diagram elements.

We can now carry out computations analogously as one does for knot diagrams,
e.g. computing that

$q^{1/4}R-q^{-1/4}R^{-1}=(q^{1/2}-q^{-1/2})I_{4},$

where $I_{4}$ is the identity matrix of dimension $4$. We conclude the
following:

######  Proposition 3.12.

Write $Q_{\mathfrak{sl}_{2}}$ to denote the assignment $\varphi$ for our
choice of $(\mathcal{C},V)$. Then this assignment constitutes the existence of
a class of triframed knotoid invariants, satisfying the properties of Figures
3.8 and 3.9.

Figure 3.9: Skein relation and other properties of $Q_{\mathfrak{sl}_{2}}$.

This class is indexed by the four free parameters $c_{1},c_{2},d_{1},d_{2}$.

Our next question is whether we can pick $c_{1},c_{2},d_{1},d_{2}$ so that
$Q_{\mathfrak{sl}_{2}}$ is invariant under the coframing identities. To this
end we calculate the value of $Q_{\mathfrak{sl}_{2}}$ on coframing loops
around either end-points. See Figure 3.10.

Figure 3.10: Value of $Q_{\mathfrak{sl}_{2}}$ on coframing loops.

If we take $c_{1}=d_{1}$ and $c_{2}=d_{2}$ then adding a $(+1)$-coframing loop
clearly has the same effect when added around either end-point, and similarly
for $(-1)$-coframing loops. Thus in this case $Q_{\mathfrak{sl}_{2}}$ is seen
to be invariant under the coframing identities.

In particular taking $c_{1}=c_{2}=d_{1}=d_{2}=1/\sqrt{2}$ we conclude:

######  Proposition 3.13.

There exists an invariant of biframed planar knotoids, denoted
$\overline{Q}_{\mathfrak{sl}_{2}}$, possessing the properties from Figure 3.9
and satisfying

$\overline{Q}_{\mathfrak{sl}_{2}}\Big{(}\raisebox{-12.80365pt}{\includegraphics[width=31.2982pt]{unknotoid.pdf}}\Big{)}=1.$

One can adjust this invariant using the integer $\text{fr}(K)$ to obtain an
$R1$-invariant version of $\overline{Q}_{\mathfrak{sl}_{2}}$ exactly as one
does for knots. For knots this results in the Jones polynomial [10, Ch. 9.4].
Using the integer $\text{cofr}(K)$ one can similarly adjust
$\overline{Q}_{\mathfrak{sl}_{2}}$ to be coframing-invariant, resulting in an
extension of the Jones polynomial to knotoids. Such an extension was already
given in [8], and these extensions must agree up to normalization since a
knotoid invariant is uniquely defined by its Skein relations and its values on
unknotted components (given that an invariant with these Skein relation
exists, which is the statement of Proposition 3.11). Also note that the
Kauffman bracket was already extended to knotoids in [15].

####  3.3.2 Option 2

Our second suggestion for extending $\varphi$ loses the advantage of being
particularly simple, but in return has several pleasant features: it is
guaranteed to be a biframed spherical knotoid invariant and requires no
additional choices after $(\mathcal{C},V)$ are fixed. Moreover it has a strong
topological motivation; namely we extend $\varphi$ by not just considering a
biframed knotoid, but a planar projection of the entire theta-curve associated
to it:

######  Definition 3.14.

Let $B$ be a braided group in a rigid ribbon category $C$. Let $K$ be a
biframed knotoid, assuming for simplicity that $K$ has integer coframing. Take
a diagram of $K$ such that the tangent vectors of $K$ at the endpoints are
vertically downwards. Now replace the leg of $K$ by a splitting $\Delta$
representing the coproduct of $B$, attaching $K$ on the left-hand side, and
replace the head by a merging $\cdot$ representing the product of $B$, again
attaching $K$ on the left. See figure 3.11.

Figure 3.11: Attaching $\Delta$ and $\cdot$ to the leg and head of $K$,
respectively. The dotted lines depict 4 open ends.

The additions $\Delta,\cdot$ now have 4 open ends: two on the right-hand side
and two vertically outgoing ends. Attach the right-hand ends via a line
segment going under any strand of $K$ it encounters. Extend the outgoing ends
vertically, going over all of $K$. Take the closure of the resulting decorated
tangle to obtain a decorated tangle $\overline{K}$. Define
$\varphi_{\theta}(K)$ to be the braided group extension of $\varphi$ evaluated
on $\overline{K}$.

######  Example 3.15.

The construction of $\varphi_{\theta}(K)$ is depicted for an example knotoid
in Figure 3.12.

Figure 3.12: The construction of $\varphi_{\theta}(K)$. The knotoid $K$ is
depicted in black.

######  Theorem 3.16.

The map $\varphi_{\theta}(K)\in\operatorname{Hom}_{\mathcal{C}}(k,k)$ is a
biframed knotoid invariant, but it is not invariant under the forbidden moves
from Figure 1.1.

###### Proof.

Invariance under $R1^{\prime},R2,R3$ and isotopies away from the end-points
follows from the invariance of $\varphi$ under these moves, so for the first
statement it suffices to prove invariance under the spherical move $R4$ and
the coframing identities.

Invariance under $R4$ is clear, since strands can be moved over
$(\cdot,\Delta)$ and hence all of $\overline{K}$, and oppositely oriented
$R1$-loops cancel by Figure 2.5.

To show invariance under the first coframing identity from Figure 2.12 we do
some preliminary computations, depicted in Figures 3.13 and 3.14.

Figure 3.13: Turning coframing into framing at the leg of $K$. Figure 3.14:
Turning coframing into framing at the head of $K$.

In Figures 3.13 and 3.14 we turn $(\cdot,\Delta)$ around to depict morphisms
$(\Delta^{*},\cdot^{*})$ on $B^{*}$. The morphisms defined as such
diagrammatically extend to a canonical braided group structure on $B^{*}$ for
which the diagrammatic moves we have used hold by definition; see [11, Prop.
4.11].

Using these preliminaries, as well as two applications of $R1^{\prime}$,
invariance of $\varphi_{\theta}(K)$ follows from the sequence of moves
depicted in Figure 3.15.

Figure 3.15: Invariance of $\overline{K}$ under the first coframing identity.

Invariance under the other coframing identity is shown analogously. For the
final statement of the theorem, let $D_{1}$ and $D_{2}$ denote the oriented
tangle diagrams that $\varphi_{\theta}$ assigns to both sides of one of the
moves in Figure 1.1. Then clearly $D_{1}$ and $D_{2}$ differ by at least one
crossing change, and so in general $\varphi_{\theta}(K)$ cannot be invariant
under either of these moves. ∎

######  Remark 3.17.

The proof of Theorem 3.16 shows in particular how $\varphi_{\theta}(K)$
changes under adjustment of the coframing: namely by adding an $R1$-loop to
the strand of $\overline{K}$ representing $\overline{e}_{+}$, and an
oppositely oriented $R1$-loop to the strand representing $\overline{e}_{-}$.

This remark also motivates why biframed knotoids are a natural object to
consider if one wants to extend quantum knot invariants to knotoids or their
associated theta-curves. Namely because quantum invariants are only
$R1^{\prime}$-invariant and not $R1$-invariant, one would expect the lines
representing $\overline{e}_{\pm}$ to need to be framed as well. As the proof
of Theorem 3.16 shows, it turns out biframed knotoids provide a proper
framework for this in the case of spherical knotoids.

Although we have assumed $K$ has integer coframing in Definition 3.14, the
above generalizes readily to half-integer coframing. The only adjustment that
needs to be made is that $\Delta$ must be replaced by $\cdot^{*}$ (or $\cdot$
by $\Delta^{*}$) at one endpoint. The construction is depicted (on its side)
for an example half-integer coframing knotoid in Figure 3.16.

Figure 3.16: Construction of $\overline{K}$ for a half-integer-coframed
knotoid (rotated $90^{\circ}$ counter-clockwise).

As a closing remark, we briefly discuss the existence of the objects required
to construct an invariant of the type suggested in Definition 3.14. To see
this existence we must exemplify braided groups in ribbon categories. Taking
the ribbon category ${}_{H}\mathcal{M}_{F}$ for a Hopf algebra $H$ and noting
that braided groups in ${}_{H}\mathcal{M}_{F}$ are Hopf algebra objects, it is
natural to hope $H$ might be a braided group, or at least a bialgebra object
in ${}_{H}\mathcal{M}_{F}$, when acting on itself suitably.

In fact if we take the adjoint action $\operatorname{Ad}$ of $H$ on itself,
then $\cdot$ is automatically an intertwiner [13, Ch. 2]. If moreover $H$ is
cocommutative then $\Delta$ is also easily seen to be an intertwiner, giving
an example of a nontrivial object suitable to apply Definition 3.14.

Acknowledgements: My gratitude goes out to Agnese Barbensi, Ulrike Tillmann,
Daniele Celoria, Jo Ellis-Monaghan, and Roland van der Veen for their advice
and guidance.

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