# The Virtual Spectrum of Linkoids and Open Curves in 3-space

Kasturi Barkataki  Louis H. Kauffman  Eleni Panagiotou

###### Abstract

The entanglement of open curves in 3-space appears in many physical systems
and affects their material properties and function. A new framework in knot
theory was introduced recently, that enables to characterize the complexity of
collections of open curves in 3-space using the theory of knotoids and
linkoids, which are equivalence classes of diagrams with open arcs. In this
paper, new invariants of linkoids are introduced via a surjective map between
linkoids and virtual knots. This leads to a new collection of strong
invariants of linkoids that are independent of any given virtual closure. This
gives rise to a collection of novel measures of entanglement of open curves in
3-space, which are continuous functions of the curve coordinates and tend to
their corresponding classical invariants when the endpoints of the curves tend
to coincide.

Keywords: linkoids, knotoids, virtual knots, open curves, topological
invariants MSC: $57M25$

##  1 Introduction

The entanglement of open curves in 3-space is a common problem of interest in
many physical systems, such as polymers, textiles and crystals [1, 2, 3]. Even
though the study of topological complexity of simple closed curves in space is
studied thoroughly in knot theory, characterizing the complexity of open
curves in 3-space has been a long-standing open problem in mathematics. A
reason is that, the traditional topological invariance ideas are not
applicable in the context of open curves in 3-space. Recent works have
introduced a framework upon which knot theory can be extended to study open
curves in 3-space without any approximation schemes [4, 5, 6]. In this paper,
new measures of entanglement of collections of open curves in 3-space are
introduced via a map between linkoids and virtual knots/links.

A knot (or link) consists in a collection of closed curve(s) embedded in three
dimensional space. Knots and links are classified, with respect to their
complexity, by topological invariants, usually of the form of integer valued
functions or polynomials with integer coefficients [7, 8, 9, 10, 11, 12]. A
topological invariant is a function on the space of knots or links that is
invariant under continuous deformations of the embeddings that do not allow
self intersections.

Knotoids/linkoids are open ended knot diagrams which can be classified under a
diagrammatic notion of equivalence similar to that of knots, but with some
restrictions regarding the isotopy moves on the endpoints of the diagram [13,
14]. As mentioned in [13], the study of knotoids is closely related to that of
virtual knots. A virtual knot diagram is one with some of the crossings being
of a different nature, called virtual and can be classified using a
diagrammatic theory which is very similar to the handling of classical
knot/link diagrams [15]. There exist many invariants of knotoids, several of
them via virtual knots, such as height, odd writhe , the Jones polynomial, the
parity bracket polynomial, the affine index polynomial and the arrow
polynomial. Even though it is often assumed that invariants of linkoids would
follow directly from those of knotoids, as is the case for knots and links, it
has been substantially harder to define invariants of linkoids.

In this paper, we give a general connection between knotoids/linkoids and
virtual links. This leads to new invariants of linkoids via virtual closure,
namely - height, genus, odd writhe, Jones polynomial, arrow polynomial and
affine-index polynomial. By accounting for all different virtual closures, new
invariants of linkoids are defined - the virtual spectrum and average spectral
invariants, which are stronger and are independent of any given virtual
closure.

Although open curves in 3-space are not knotted or linked in the topological
sense, they can form complex conformations, which are called entangled.
Entanglement of open curves can also be measured via topological/geometrical
measures, which are either real numbers or polynomials with real coefficients,
that are continuous functions of the curve coordinates and can detect knotting
and threading [6, 16, 4, 5]. A projection of a collection of curves in 3-space
can be a closed or open knot/link diagram. In [4, 6] it was shown that a
rigorous measure of complexity of a collection of open curves in 3-space is
given by taking the average of an invariant of a projection of the curves over
all possible directions of projection.

In this paper, the special construction of the virtual closure of open curves
in 3-space is introduced, which leads to the definition of new measures of
entanglement of collections of open curves in 3-space. These are continuous
functions of the curve coordinates and tend to the classical topological
invariant as the endpoints coincide.

The paper is organised as follows : Section 2 gives an exposition into some
basic definitions and properties associated with virtual knots. Section 3
introduces the generalized virtual closure of linkoids and makes a rigorous
connection between the theory of linkoids and the theory of virtual knots.
Section 4 discusses the emergence of new invariants of linkoids via the
virtual closure introduced in Section 3. Section 5 discusses the virtual
spectrum of linkoids and new invariants of linkoids independent of any given
virtual closure. Section 6 introduces the virtual closure of open curves in
3-space and new measures of entanglement of open curves in 3-space and
discusses their properties. Section 7 presents the conclusions of this study.

##  2 Basics on Virtual Knots and Links

This section gives an overview of the basic definitions pertaining to virtual
knots and links and their properties [15, 14, 17, 18].

######  Definition 2.1.

(Virtual knot/link and virtual knot/link diagram) A virtual knot/link diagram
consists of generic closed curves in $\mathbb{R}^{2}$ (or $S^{2}$) such that
each crossing is either a classical crossing with over and under arcs, or a
virtual crossing without over or under information. Virtual knot/link diagrams
are classified using the generalized Reidemeister moves, which include the
classical Reidemeister moves and the virtual Reidemeister moves (See Figure 1)
and the forbidden moves (See Figure 2). A virtual knot/link is defined as an
equivalence class of virtual knot/link diagrams under the generalised
Reidemeister moves. In conjuction with the generalized Reidemeister moves, a
segment of a diagram, consisting of a sequence of consecutive virtual
crossings, can be excised and a new connection be made between the resulting
free ends. If the new connecting segment intersects the remaining diagram
(transversally) then each new intersection is taken to be virtual. Such an
excision and reconnection is called a detour move (See Figure 2).

Figure 1: Generalised Reidemeister moves on virtual knots/links : the
classical Reidemeister moves, $R_{1}$, $R_{2}$, $R_{3}$; the virtual moves,
$V_{1}$, $V_{2}$, $V_{3}$; and the semi-virtual move $SV$.

Figure 2: Forbidden moves on virtual knots/links : the move involving an over-
strand, FO and the move involving an under-strand, FU; and the detour move of
an arc in a virtual knot diagram.

Given a virtual knot/link diagram, $VL_{g}$, with $g$ virtual crossings, there
is a canonical way to associate it with a knot/link in a thickened surface :
Each virtual crossing in the diagram can be considered as a shorthand for a
detour of one of the arcs involved in the crossing through a 1-handle that has
been attached to the 2-sphere of the original diagram. The two choices (above
or below) for the 1-handle detour are homeomorphic to each other (as abstract
manifolds with boundary). This gives an embedding (the canonical embedding) of
a collection of circles into the sphere with $g$ handles. Any other embedding,
which is stably equivalent to the canonical embedding, is considered to be a
valid embedding of $VL_{g}$. In fact, two virtual knot/link diagrams are
equivalent if and only if their corresponding surface embeddings are stably
equivalent [15, 17].

Figure 3: (Left) A diagram of a virtual trefoil knot, (Center) Its ribbon
neighborhood representation, (Right) Its surface representation.

######  Remark 2.1.

A stably equivalent way to realize virtual knots is to form a ribbon
neighborhood surface/abstract link diagram for the given virtual knot or link
diagram [19]. In such a diagram, the classical crossings are represented as
diagrammatic crossings in disks, which are connected by ribbons, while the
virtual crossings are represented by crossings between two ribbons that pass
over/under one another without interacting. The ribbon neighborhood diagram is
independent of the choice of over or under information for the ribbon strips.
By gluing discs to the boundary circles of a ribbon neighborhood diagram of a
virtual knot, we obtain a unique surface representation of the virtual knot
(See Figure 3).

The virtual crossing number of a virtual knot is defined as follows [18, 20]:

######  Definition 2.2.

(Virtual crossing number, and Minimal virtual knot/link diagram) The virtual
crossing number, $\mathsf{cr}_{v}(VL)$ of a virtual knot/link $VL$ is the
minimum of the number of virtual crossings over all diagrams of $VL$. A
diagram corresponding to a virtual knot/link $VL$ is said to be minimal if the
number of virtual crossings in the diagram is equal to $\mathsf{cr}_{v}(VL)$.
The minimal diagram of a virtual knot/link is unique upto classical
Reidemeister moves.

A virtual knot/link can also be studied as a knot in a thickened surface
through its minimal representation [21, 22], defined as follows:

######  Definition 2.3.

(Genus of a virtual knot/link and Minimal representation of a virtual
knot/link) The genus, $g$, of a virtual knot/link is defined as the genus of
its minimal embedding surface. An embedding of a virtual knot/link is said to
be its minimal representation if it is an embedding in an abstract surface,
$S$, whose genus is equal to the genus of the given virtual knot. The minimal
representation of a virtual knot/link is unique up to homeomorphism of the
surface $S$.

######  Remark 2.2.

Let $D(V)$ be a diagram of a virtual knot, $V$ and $a(D(V))$ be its ribbon
neighborhood diagram. The surface representation of $V$, obtained by gluing
discs to the two boundary circles in $a(D(V))$, realizes the unique (up to
homeomorphism) minimal embedding surface of $V$.

##  3 Linkoids and Virtual Links

In this section an analogy is drawn between virtual knots/links and virtual
closures of knotoids/linkoids. Linkoids have been typically studied as
diagrammatic objects and they can be thought of as projections of open curves
in 3-space [6, 14, 13, 23, 24, 25]. Throughout this section, the symbol
$\Sigma$ is used to denote a surface on which a linkoid diagram lies. The
results can be generalized for any $\Sigma$ but in this manuscript, it is
assumed that $\Sigma=S^{2}=\mathbb{R}^{2}\cup\infty$.

######  Definition 3.1.

(Knotoid/linkoid diagram and Knotoid/linkoid) A knotoid/linkoid diagram $L$
with $n\in\mathbb{N}$ components in $\Sigma$ is a generic immersion of
$\bigsqcup_{i=1}^{n}[0,1]$ in the interior of $\Sigma$ whose only
singularities are transversal double points endowed with over/undercrossing
data. These double points are called the crossings of $L$. The immersion of
each $[0,1]$ is referred to as a component of the knotoid/linkoid diagram and
the images of $0$ and $1$ under this immersion are called the foot and the
head of the component, respectively. These two points are distinct from each
other and from the double points; they are called the endpoints of the
component. The diagram $L$ has a total of $2n$ endpoints. A knotoid/linkoid is
an equivalence class of knotoid/linkoid diagrams up to the equivalence
relation induced by the three Reidemeister moves and isotopy. It is forbidden
to pull the strand adjacent to an endpoint over/under a transversal strand
(See Figure 4).

Figure 4: Omega moves (Reidemeister moves) and forbidden moves
($\Phi_{+},\Phi_{-}$) on linkoid diagrams.

Given a linkoid $L$ with $n$ components, a choice of labelling can be made for
the $2n$ endpoints with numbers from the set $\\{1,2,\cdots,2n\\}$, without
repetition. The initial connectivity among endpoints of $L$ allows us to
define a strand permutation as follows :

######  Definition 3.2.

(Strand Permutation) Let the endpoints of a linkoid diagram, $L$, with $n$
components, be denoted by labels from the set $E=\\{1,2,\cdots,2n\\}$, without
repetition. The strand permutation of $L$ is defined to be the element,
$\tau\in S_{2n}$, such that, for any $i\in E$, i.e. the head/foot of a
component, $\tau(i)\in E$ is the corresponding foot/head and $\tau(i)\neq i$.
It follows that $\tau$ is of order $2$. In other words, $i$ and $\tau(i)$ are
labels for the two endpoints of an open component (strand) in the linkoid.

Figure 5: (Above) The labelled endpoints of any linkoid determine a strand
permutation. Examples of a tangle, a braid and a general linkoid are shown
above with strand permutations $(1\quad 3)(2\quad 4)$, $(1\quad 6)(3\quad
4)(2\quad 5)$ and $(1\quad 3)(2\quad 4)$, respectively. (Below) The virtual
closure of the tangle, the braid and the linkoid according to their respective
strand permutations.

By definition, the strand permutation for a linkoid with $n$ components, is a
product of $n$ disjoint transpositions (See Figure 5). In the following, given
a permutation $\tau$ in $E=\\{1,2,\cdots,2n\\}$, the notation $L_{\tau}$ is
used to represent a labelled linkoid with $n$ components with the head-foot
labelling implied by its strand permutation, $\tau$.

###  3.1 Generalized Virtual Closure of Linkoids

In this section, the relationship between multi-component linkoids and virtual
links is examined for the first time and a general framework for studying the
map between linkoids and virtual knots/links is introduced. The relationship
between knotoids (single component) and virtual knots has been explored before
[13, 14]. It was shown that, by introducing a closure arc to a knotoid
diagram, which only creates virtual crossings, one can obtain a virtual knot
diagram. The extension of the approach presented in [13, 14] to the case of
linkoids with more than 1 component, is substantially more complex and it is
studied for the first time in this manuscript.

Indeed, notice that, there are multiple ways to close a linkoid of more than 2
components and it is possible to obtain non-equivalent virtual knots/links,
for the same linkoid, based on the choice of closure of the endpoints. To make
this precise, the following definition of a closure permutation will be useful
:

######  Definition 3.3.

(Closure Permutation) Let the endpoints of a linkoid diagram, $L$, with $n$
components, be denoted by labels from the set $E=\\{1,2,\cdots,2n\\}$, without
repetition. A closure permutation of $L_{\tau}$, where $\tau$ denotes the
strand permutation of $L$, is any element, $\sigma\in S_{2n}$, such that,
$\sigma^{2}=\operatorname{id}$ and $\sigma(i)\neq i$, for all $i\in E$.

######  Remark 3.1.

Definitions 3.2 and 3.3 imply the following : (i) The strand permutation is a
special case of a closure permutation. (ii) Any closure permutation can be
expressed as the product of $n$ disjoint transpositions. (iii) Strand
permutation (resp. closure permutation) of a linkoid is synonymous with the
term head-foot pairing (resp. pairing combination) introduced in [6].

Figure 6: The virtual closure of a linkoid can give rise to non-equivalent
virtual knots depending on the choice of the closure permutation, as seen in
the above example. (Left) An example of a linkoid, $L$, with $\tau=(1\quad
2)(3\quad 4)$. (Center) The virtual closure, $(L_{\tau},\tau)$ of $L$, with
respect to $\tau$, gives rise to a virtual link. It is in fact the strand
closure of $L$. (Right) Whereas, the virtual closure, $(L_{\tau},\sigma)$ with
respect to another closure permutation, $\sigma=(1\quad 3)(4\quad 2)$, gives
rise to a trefoil.

Any virtual closure of a linkoid diagram, with respect to a given closure
permutation, can be defined as follows:

######  Definition 3.4.

(Virtual Closure of a linkoid diagram with respect to $\sigma$ and Strand
Closure) Let $L_{\tau}$ be a linkoid diagram with $n$ components; $\tau$ be
its strand permutation and $\sigma\in S_{2n}$ be a closure permutation of
$L_{\tau}$. The virtual closure of $L_{\tau}$ with respect to $\sigma$,
denoted $(L_{\tau},\sigma)_{v}$, is defined to be the virtual knot/link
obtained by introducing a closure arc between $i$ and $\sigma(i)$, for every
$i\in\\{1,2,\cdots,2n\\}$, consisting entirely of virtual crossings. The
special case of the virtual closure with $\sigma=\tau$ will be called the
strand closure of $L$. (See Figure 6).

Depending on the number of open-ended arcs, the strand permutation and the
choice of closure permutation, the number of components of a virtual closure
of a linkoid can be determined by its associated linkoid and closure
permutations, by generalizing the notion of segement cycle (introduced in
[6]), as follows:

######  Definition 3.5.

(Segment cycle of a linkoid with respect to a closure permutation) Let
$L_{\tau}$ be a linkoid with $n$ components, with strand permutation $\tau$
and a closure permutation $\sigma$. Let $E=\\{1,2,\cdots,2n\\}$ denote the set
of the labelled endpoints of $L_{\tau}$. A segment cycle of $L_{\tau}$ with
respect to the closure permutation $\sigma$, is defined to be an orbit of a
point in $E$, under the action of the group $\langle\tau,\sigma\rangle$ on
$E$.

Figure 7: (Left) A linkoid, $L$, with 2 components and strand permutation,
$\tau=(1\quad 3)(2\quad 4)$; (Center) The virtual closure,
$(L_{\tau},\sigma)_{v}$, with respect to $\sigma=(1\quad 4)(2\quad 3)$;
(Right) the unique segment cycle in $L$, with respect to $\sigma$.

The definition of a segment cycle and the defining properties of a group
guarantee that the set of segment cycles of (points in) $E$ under the action
of $\langle\tau,\sigma\rangle$ form a partition of $E$. The set of all segment
cycles of the linkoid under the action of $\langle\tau,\sigma\rangle$ is
written as the quotient of the action, i.e, $\displaystyle
E/\langle\tau,\sigma\rangle$. The number of components in a closure of a
linkoid can be detremined by the segment cycles as follows :

######  Proposition 3.1.

Let $L_{\tau}$ be a linkoid with $n$ components, with strand permutation
$\tau$ and a closure permutation $\sigma$. Let $E=\\{1,2,\cdots,2n\\}$ denote
the set of the labelled endpoints of $L_{\tau}$. Let
$G=\langle\tau,\sigma\mid\tau^{2}=\sigma^{2}=1\rangle$. The number of distinct
segment cycles, $\displaystyle|E/G|$ of $L_{\tau}$, with respect to $\sigma$
is given as

$\displaystyle|E/G|=\frac{1}{2m}\sum_{g\in G}|E^{g}|$

where $m=|\tau\sigma|$ is the order of $\tau\sigma\in G$ and $E^{g}$ denotes
the set of elements in $E$ that are fixed by $g\in G$ i.e. $\displaystyle
E^{g}=\\{x\in E\quad|\quad g.x=x\\}$.

###### Proof.

$G$ can alternatively be expressed as,
$\langle\tau\sigma,\sigma\mid(\tau\sigma)^{m}=\sigma^{2}=(\sigma\tau\sigma)^{2}=1\rangle$.
Thus, $G$ is isomorphic to the dihedral group $D_{m}$. By Definition 3.5,
segment cycles of $E$ form a partition of $E$ under the action of $G$.
Therefore, by Burnside’s counting theorem, the number of distinct segment
cycles of $L_{\tau}$ with respect to the closure permutation, $\sigma$ is
given as,

$\displaystyle|E/G|=\frac{1}{|G|}\sum_{g\in
G}|E^{g}|=\frac{1}{2m}\sum_{g\in\langle\tau,\sigma\rangle}|E^{g}|.$ (1)

∎

######  Corollary 3.1.

Let $L_{\tau}$ be a linkoid with strand permutation $\tau$ and a closure
permutation $\sigma$. The number of components in the virtual closure
$(L_{\tau},\sigma)_{v}$ is equal to the number of segment cycles of $L_{\tau}$
with respect to $\sigma$.

###### Proof.

Let $E$ be the set of all labelled endpoints of the linkoid, $L_{\tau}$. If
$i\neq j\in E$ lie in the same component, then $i,j$ are either in the same
linkoid component or they belong to different components that are connected
via closure, thus they are in the same orbit under the action of the group
$\langle\tau,\sigma\rangle$ on $E$, which is the same segment cycle.
Similarly, a segment cycle defines a component of the virtual closure, since,
if $i,j$ are in different components (different linkoid components and not
connected via closure), they are not related by the action of
$\langle\tau,\sigma\rangle$ on $E$ (See Figure 7). ∎

######  Remark 3.2.

Braids and tangles are special types of linkoids. Hence, Definitions 3.1-3.4,
Proposition 3.1 and Corollary 3.1 apply naturally to braids and tangles. Given
any braid, $B$, with $n$ components, is it interesting to consider the two
very special kinds of braid closures - the standard braid closure and the
strand closure. As a convention, the endpoints of a braid are numbered, in
order, from the top bar to the bottom bar. The standard braid closure is
obtained by introducing a closure arc between $i$ and $i+n$, for all
$i\in\\{1,2,\ldots,n\\}$. On the other hand, the strand closure is obtained by
introducing a closure arc between $i$ and $\tau(i)$, for all
$i\in\\{1,2,\ldots,2n\\}$, where $\tau$ is the strand permutation associated
with the braid (See Figure 8).

Figure 8: (Left) Standard braid closure and (Right) Strand closure of a braid
on 3 strands.

######  Proposition 3.2.

Let $L$ be a linkoid diagram with strand permutation, $\tau$, and a closure
permutation, $\sigma$. The virtual closure, $(L_{\tau},\sigma)_{v}$, is
independent of the immersion class of the closure arcs.

###### Proof.

Let $A$ and $B$ be two diagrams, of the virtual closure of the given linkoid,
with different immersion classes of the closure arcs. The diagram $B$ can be
obtained by detour moves on the closure arcs in $A$. ∎

######  Remark 3.3.

For a linkoid, $L_{\tau}$, the virtual closure with respect to the strand
permutation, $(L_{\tau},\tau)$, corresponds to the component-wise closure of
$L_{\tau}$. Thus, the virtual closure of a linkoid corresponding to the strand
permutation (i.e. $\sigma=\tau$) has the same number of components as the
linkoid.

######  Proposition 3.3.

Let $L_{\tau}^{1},L_{\tau}^{2}$ denote two equivalent linkoid diagrams with
strand permutation $\tau$ and let $\sigma$ denote a closure permutation. Then
their virtual closures with respect to $\sigma$, $(L_{\tau}^{1},\sigma)$ and
$(L_{\tau}^{2},\sigma)$ are equivalent.

###### Proof.

It follows by performing a sequence of Reidemeister moves on the virtual
knot/link, $(L_{\tau}^{1},\sigma)$, which transforms $L_{\tau}^{1}$ to
$L_{\tau}^{2}$. ∎

######  Remark 3.4.

Let $L$ be a linkoid diagram with strand permutation, $\tau$, and a closure
permutation, $\sigma$ and let $(L_{\tau},\sigma)_{v}$ be the corresponding
unique virtual knot/link obtained via virtual closure. $(L_{\tau},\sigma)_{v}$
can be interpreted in terms of links in thickened surfaces or on ribbon-
neighborhoods (See Section 2).

######  Definition 3.6.

(Minimal virtual closure and Minimal virtual representative of a linkoid
diagram with respect $\sigma$) A virtual closure of a linkoid diagram,
$L_{\tau}$, with respect to a closure permutation, $\sigma$, is said to be
minimal if no virtual crossing can be further removed via a sequence of
generalized Reidemeister moves and the 0-move on $S^{2}$. The resultant
virtual knot/link, $(L_{\tau},\sigma)_{v}$, is called the minimal virtual
representative of $L_{\tau}$ with respect to $\sigma$.

######  Definition 3.7.

(Link-type linkoid with respect to $\sigma$) A linkoid $L_{\tau}$ is said to
be link-type with respect to a closure permutation, $\sigma\in S_{2n}$, if the
minimal representative diagram of $(L_{\tau},\sigma)_{v}$ has no virtual
crossings. Thus, the virtual closure of $L_{\tau}$, with respect to $\sigma$,
gives rise to a classical link, $\mathcal{C}$. Then, the link-type linkoid,
$L_{\tau}$, is said to be of link-type $\mathcal{C}$ with respect to
$\sigma\in S_{2n}$.

###  3.2 The Virtual Closure Map

The virtual closures of linkoids define a map between linkoids and virtual
links which can be formalized by the means of the virtual closure map, as
follows:

######  Definition 3.8.

(Virtual closure map) Let for every $n\in\mathbb{N}$, $\mathcal{A}_{n}$ be the
set of all linkoids with $n$ components and $H_{n}=\\{\sigma\in
S_{2n}:\sigma^{2}(i)=i,\sigma(i)\neq i,\quad\forall i\in E\\}$, where
$E=\\{1,2,\ldots,2n\\}$. Let $\mathcal{V}$ be the set of all virtual
knots/links. The virtual closure map, $\varphi$, is defined as,

$\begin{split}\displaystyle\varphi&:\quad\\{(L,\tau,\sigma)\mid(L,\tau,\sigma)\in\mathcal{A}_{n}\times
H_{n}\times H_{n}\text{ for some
}n\in\mathbb{N}\\}\longrightarrow\mathcal{V}\\\
&(L,\tau,\sigma)\mapsto(L_{\tau},\sigma)_{v},\quad\text{where
}L\in\mathcal{A}_{n}\text{ and }\tau,\sigma\in H_{n}.\\\ \end{split}$

where, $(L_{\tau},\sigma)_{v}$ denotes the virtual closure, with respect to
the closure permutation, $\sigma$, of the linkoid, $L$, with strand
permutation, $\tau$.

Figure 9: (Left and Right) Two non-equivalent knotoid diagrams, $L_{1}$ and
$M_{1}$, respectively. (Center) The virtual closures of $L_{1}$ and $M_{1}$
are equivalent and they correspond to the virtual trefoil knot.

######  Theorem 3.1.

The virtual closure map, $\varphi$, as given in Definition 3.8, is surjective
but not injective.

###### Proof.

Let $L_{1}$ and $M_{1}$ be the two knotoids given in Figure 9 and for all
$n\geq 2$, let,

$\displaystyle L_{n}=L_{1}\hskip 3.0pt\bigsqcup_{i=2}^{n}\hskip
5.0pt\raisebox{-13.0pt}{\includegraphics[scale={0.35}]{fig/triv-1-strand.png}}_{2i}^{2i-1}$
and $\displaystyle M_{n}=M_{1}\hskip 3.0pt\bigsqcup_{i=2}^{n}\hskip
3.0pt\raisebox{-13.0pt}{\includegraphics[scale={0.35}]{fig/triv-1-strand.png}}_{2i}^{2i-1}$

where
$\raisebox{-11.0pt}{\includegraphics[scale={0.3}]{fig/triv-1-strand.png}}_{2i}^{2i-1}$
is a crossingless open segment with endpoints $2i$ and $2i-1$, for all $2\leq
i\leq n$. Clearly, $L_{n}$ and $M_{n}$ are not equivalent as linkoids. For
both $L_{n},M_{n}\in\mathcal{A}_{n}$, let the strand permutation, $\tau$ and
the closure permuataton $\sigma$ be each equal to $\prod_{i=1}^{n}(2i-1\quad
2i)$. The map $\varphi$ is not injective since,

$\varphi(L_{n},\tau,\sigma)=\raisebox{-13.0pt}{\includegraphics[scale={0.4}]{fig/v-tref-
closed.png}}\hskip 3.0pt\bigsqcup_{i=2}^{n}\hskip
3.0pt\bigcirc=\varphi(M_{n},\tau,\sigma).$

For any $V\in\mathcal{V}$, let $DV$ be a diagram with $m$ virtual crossings.
For each virtual crossing, a unique arc may be chosen which passes through the
virtual crossing but does not interact with the rest of $DV$. These $m$ arcs
may be labelled as $(2i-1\quad 2i)$, where $1\leq i\leq m$. Let $L$ be the
linkoid diagram with $m$ components which is identical to $DV$ but does not
contain the arcs bounded by $(2i-1\quad 2i)$. Clearly, $DV$ is the virtual
closure of $L$ with respect to $\sigma=\prod_{i=1}^{n}(2i-1\quad 2i)$. Hence,
$\varphi$ is surjective.

∎

The virtual closure map has the following properties :

  1. 1.

The image of the restriction map,
${\varphi}\big{|}_{L,\tau}:H_{n}\longrightarrow\mathcal{V}$, where $n$ is the
number of components in $L$ is a subset of $\mathcal{V}$ and is denoted as
$\displaystyle\operatorname{Im}({\varphi}{\big{|}}_{L,\tau})$. The number of
elements in this subset is equal to
$\displaystyle{\big{|}}\operatorname{Im}({\varphi}{\big{|}}_{L,\tau}){\big{|}}\leq|H_{n}|=\frac{1}{n!}\binom{2n}{2}\binom{2n-2}{2}\ldots\binom{2}{2}$.
Therefore, there can be at most $|H_{n}|$ virtual knots/links that represent
$L$ via the virtual closure map.

  2. 2.

Let $L$ be a trivial linkoid with $n$ components, and without loss of
generality, let $\tau=\prod_{i=1}^{n}(2i-1\quad 2i)$ i.e. $\displaystyle
L=\bigsqcup_{i=1}^{n}\hskip
5.0pt\raisebox{-11.0pt}{\includegraphics[scale={0.3}]{fig/triv-1-strand.png}}_{2i}^{2i-1}$.
Then, the virtual closure of $L$ with respect to any $\sigma\in H_{n}$ is a
trivial link (up to generalized Reidemeister moves) with $\leq n$ components.
In particular, virtual closure with respect to $\displaystyle\sigma=(1\quad
n)\prod_{i=1}^{n-1}(2i\quad 2i+1)$ corresponds to an unknot whereas, virtual
closure with respect to $\tau$ corresponds to a trivial link with $n$
components. Hence,
$\displaystyle{\big{|}}\operatorname{Im}({\varphi}{\big{|}}_{L,\tau}){\big{|}}=n$.

  3. 3.

Consider $\tau$ and $\sigma$ defined on $2n$ endpoints. For a given virtual
link, $V\in\mathcal{V}$, of $m$ components, the preimage of the restriction
map,
$\displaystyle{\varphi}{\big{|}}_{\tau,\sigma}:\mathcal{A}_{n}\longrightarrow\mathcal{V}$,
is the set of all $n$-component linkoids, with strand permutation $\tau$,
whose virtual closure with respect to $\sigma$ is equal to $V$.

  4. 4.

Without loss of generality, let $\tau=\prod_{i=1}^{n}(2i-1\quad 2i)$, for a
linkoid $L$ with $n$ components. If $L$ has more than 1 component,
${\big{|}}\operatorname{Im}({\varphi}{\big{|}}_{L,\tau}){\big{|}}>1$ (Since
virtual closure with respect to $\tau$ yields a virtual link with
$n$-components, whereas virtual closure with respect to $\sigma=(1\quad
n)\prod_{i=1}^{n-1}(2i\quad 2i+1)$ corresponds to a virtual knot with 1
component).

######  Remark 3.5.

Under the map in Definition 3.8, linkoids can be associated to embeddings of
links in thickened surfaces of arbitrary genus.

##  4 Invariants of Linkoids via Virtual Closure

The framework introduced in the previous sections enables to define a new
collection of linkoid invariants via invariants of virtual links.

The following are some well-known invariants of virtual knots/links:
fundamental group, quandles, biquandles, odd writhe, Jones polynomial,
Khovanov homology, arrow polynomial, index polynomial, quantum link invariants
and virtual Vassiliev invariants. All these can be defined for linkoids using
the following theorem:

######  Theorem 4.1.

Let $L_{\tau}$ be a linkoid, where $\tau$ is the strand permutation and let
$\sigma$ a closure permutation. For any $\mathcal{F}$, an invariant of virtual
links, let the map $\mathcal{F}_{\sigma}$, be defined as,

$\displaystyle{\mathcal{F}_{\sigma}}(L_{\tau}):=\mathcal{F}((L_{\tau},\sigma)_{v}),$
(2)

where $(L_{\tau},\sigma)_{v}$ is the virtual closure of $L_{\tau}$
corresponding to $\sigma$. Then $\mathcal{F}_{\sigma}$ is an invariant of
linkoids.

###### Proof.

Let $L_{\tau}$ and $K_{\tau}$ be diagrams that differ by a Reidemeister move
on linkoids. By Remark 3.2, the virtual closure diagrams,
$(L_{\tau},\sigma)_{v}$ and $(K_{\tau},\sigma)_{v}$, represent the same
virtual knot, where $\sigma$ is the given closure permutation for
$\mathsf{L}$. Thus,

$\displaystyle\mathcal{F}_{\sigma}(L_{\tau})=\mathcal{F}((L_{\tau},\sigma)_{v})=\mathcal{F}((K_{\tau},\sigma)_{v})=\mathcal{F}_{\sigma}(K_{\tau}),$

. ∎

######  Remark 4.1.

Note that the virtual class of the closure, with respect to a closure
permutation $\sigma$, of a linkoid $\mathsf{L}_{\tau}$ is by itself an
invariant of $\mathsf{L}_{\tau}$. This virtual class of the closure is denoted
$(\mathsf{L}_{\tau},\sigma)_{v}$.

Theorem 4.1 gives rise to the following corollary regarding link-type
linkoids, which plays a critical role in Section 6, which discusses
entanglement of open curves in 3-space.

######  Corollary 4.1.

Let ${L}_{\tau}$ be a link-type linkoid of type $\mathcal{C}$ (See Definition
3.7), with respect to a closure permutation $\sigma$, where $\tau$ denotes the
strand permutation. Let $\mathcal{F}$ denote an invariant of virtual links and
$\mathcal{F}_{\sigma}$ the corresponding invariant of linkoids. Then

$\displaystyle\mathcal{F}_{\sigma}(L_{\tau})=\mathcal{F}(\mathcal{C})$ (3)

###### Proof.

It follows by Definition 3.7 and Theorem 4.1.∎

The virtual closure map, $\varphi$, defined in Definition 3.8 is not
injective, where $n\in\mathbb{N}$. Therefore, linkoids which are mapped to the
same virtual knot/link under the virtual closure map, return the same value
for any invariant of linkoids as defined in Theorem 4.1.

A more detailed discussion on some specific invariants of linkoids is provided
in the remainder of this section. A combinatorial approach to determine the
Jones polynomial and the arrow polynomial of linkoids, with respect to
arbitrary closure permutations, is discussed and it is shown to be consistent
with the definition in terms of virtual closure. Moreover, the extension of
invariants such as, height, genus, odd writhe and the affine-index polynomial
is discussed for linkoids in terms of virtual closure.

###  4.1 Height and Genus of Linkoids and Virtual Links

A fundamental topological invariant of a knotoid (linkoid with one component)
is its height [13, 14]. Here, a generalized definition of the height of
linkoids is given by referring to virtual crossings and the choice of closure
permutation.

######  Definition 4.1.

(Height of a linkoid with respect to $\sigma$) The height of a linkoid diagram
$L_{\tau}$, corresponding to a closure permutation $\sigma$, is the number of
virtual crossings in its minimal virtual closure. The height of a linkoid,
$\mathsf{L}_{\tau}$ corresponding to $\sigma$ is defined as the minimum of the
diagrammatic heights, taken over all knotoid/linkoid diagrams equivalent to
$\mathsf{L}_{\tau}$.

######  Remark 4.2.

The height of a knotoid, as defined in [13, 14], is recovered as a special
case of Definition 4.1 by taking the strand closure of the knotoid.

Via virtual closure, the genus of a linkoid can be defined as follows:

######  Definition 4.2.

(Genus of linkoid with respect to $\sigma$ ) The genus, $g$, of a linkoid
$L_{\tau}$ with respect to a closure permutation, $\sigma$ is defined as the
genus of the minimal embedding surface of the virtual knot/link,
$(L_{\tau},\sigma)$.

###  4.2 The Jones polynomial of Linkoids and Virtual Links

In this section, the Jones polynomial of a knotoid/linkoid, with respect to a
given closure permutation, is discussed as the normalized bracket polynomial
of the knotoid/linkoid, with the substitution $A=t^{-1/4}$. The bracket
polynomial for linkoids was recently defined combinatorially, without
referring to its virtual closure in [6]. In this section, it is shown that the
definition in [6] coincides with that of the strand virtual closure. Moreover,
it is shown that the combinatorial definition of the bracket polynomial can be
generalized to account for arbitrary closure permutations and that it
coincides with the corresponding definition in terms of virtual closure.

Below follows a combinatorial definition of the Jones polynomial of linkoids :

######  Definition 4.3.

(Jones polynomial of a linkoid with respect to $\sigma$) The Jones polynomial
of an oriented linkoid diagram, $L$, with respect to a closure permutation,
$\sigma$, is defined as,

$f_{L^{\sigma}}=(-A^{3})^{-\mathsf{Wr}(L)}\langle L^{\sigma}\rangle\hskip
1.42271pt,$ (4)

where $\mathsf{Wr}(L)$ is the writhe of the linkoid diagram and $\langle
L^{\sigma}\rangle$ is the generalized bracket polynomial of $L$, with respect
to $\sigma$ (See Definition 4.4).

The generalized bracket polynomial in Definition 4.3 can be defined
combinatorially as follows:

######  Definition 4.4.

(Generalized bracket polynomial of a linkoid with respect to $\sigma$) Let $L$
be a linkoid diagram with $n$ components, $E$ be the set of its labelled
endpoints and $\sigma$ be a closure permutation. The generalized bracket
polynomial of $L$, with respect to $\sigma$, is completely characterised by
the following skein relation and conditions:

$\left\langle\raisebox{-7.0pt}{\includegraphics[width=17.34189pt]{fig/cross10.png}}\right\rangle=A\left\langle\raisebox{-7.0pt}{\includegraphics[width=17.34189pt]{fig/cross30.png}}\right\rangle+A^{-1}\left\langle\raisebox{-7.0pt}{\includegraphics[width=17.34189pt]{fig/cross20.png}}\right\rangle,\hskip
14.22636pt\bigl{\langle}\tilde{L}\cup\bigcirc\bigr{\rangle}=d\bigl{\langle}\tilde{L}\bigr{\rangle},\hskip
14.22636pt\left\langle
S_{\ell}\right\rangle=d^{|E/\langle\tau_{S},\sigma\rangle|}\quad,$ (5)

where $d=-A^{2}-A^{-2}$ ; $\tilde{L}$ is any linkoid diagram ; $S_{\ell}$ is
the trivial linkoid formed by the collection of $n$ labelled open segments in
a state, $S$, in the state sum expansion of $L$ and $\tau_{S}$ is the strand
permutation of $S_{\ell}$. The term, $|E/\langle\tau_{S},\sigma\rangle|$, is
the number of distinct segment cycles in $S_{\ell}$, with respect to $\sigma$.

The generalized bracket polynomial of $L$, with respect to $\sigma$, can be
expressed as the following state sum expression :

$\bigl{\langle}L^{\sigma}\bigr{\rangle}\hskip 1.42271pt:=\hskip
1.42271pt\sum_{S}A^{\alpha(S)}d^{\operatorname{circ}(S)+|E/\langle\tau_{S},\sigma\rangle|-1}\quad,$
(6)

where $S$ is a state corresponding to a choice of smoothing over all double
points in $L$; $\alpha(S)$ is the algebraic sum of the smoothing labels of $S$
and $\operatorname{circ}(S)$ is the number of circular components in $S$.

The generalized bracket polynomial of linkoids has the following properties :

  1. 1.

It is a one-variable polynomial that preserves the underlying skein relation
of the bracket polynomial of classical knots and knotoids.

  2. 2.

If $L$ is a link diagram, with no open components, then the labelled set of
endpoints, $E=\emptyset$ and the generalized bracket polynomial coincides with
the traditional Kauffman bracket polynomial of $L$ i.e. $\displaystyle\langle
L\rangle=\sum_{S}A^{\alpha(S)}d^{\operatorname{circ}(S)-1}\hskip 1.42271pt.$

  3. 3.

If $L$ is a knotoid diagram (linkoid with one component), then the generalized
bracket polynomial coincides with the Kauffman bracket polynomial of knotoids.
Any closure permutation connects the 2 open endpoints of $L$, thus
$|E/\langle\tau_{S},\sigma\rangle|=1$ for all $S$. Thus, $\displaystyle\langle
L\rangle=\sum_{S}A^{\alpha(S)}d^{\operatorname{circ}(S)}\hskip 1.42271pt.$

  4. 4.

For an $n$-component linkoid diagram, $L$, with strand permutation, $\tau$,
and no crossings, the state sum only has 1 state, $S$. In this case,
$\tau_{S}=\tau$ and $\displaystyle\langle
L^{\sigma}\rangle=d^{|E/\langle\tau,\sigma\rangle|-1}\hskip 1.42271pt$, where
$E$ denotes the set of labelled endpoints of $L$.

  5. 5.

If a linkoid diagram, $L$, is of link-type, with respect to a closure
permutation, $\sigma$, the bracket polynomial coincides with that of the
corresponding link upon the closure of endpoints.

According to Theorem 4.1, the Jones polynomial of a linkoid, $L_{\tau}$, with
respect to a closure permutation $\sigma$, can be defined as the Jones
polynomial of its virtual closure, $(L_{\tau},\sigma)_{v}$. Theorem 4.2 shows
the equivalence between Definition 4.3 and the definition of the Jones
polynomial via virtual closure.

######  Theorem 4.2.

Let $L$ be a linkoid diagram with strand permutation, $\tau$ and closure
permutation, $\sigma$ on the labelled set of endpoints, $E$. The Jones
polynomial of $L$, according to Definition 4.3, is equal to the Jones
polynomial of the virtual closure, $(L_{\tau},\sigma)_{v}$, namely,
$f_{L^{\sigma}}=f_{(L_{\tau},\sigma)_{v}}$.

###### Proof.

$L$ and $(L_{\tau},\sigma)_{v}$ have equal number of classical crossings,
therefore $\mathsf{Wr}(L)=\mathsf{Wr}\left((L_{\tau},\sigma)_{v}\right)$. Let
$S$ be a state in the state sum expansion of $L$ corresponding to a choice of
smoothing over the classical crossings in $L$. By the one-to-one
correspondence between the classical crossings in $L$ and
$(L_{\tau},\sigma)_{v}$, there is a state $S_{v}$ in $(L_{\tau},\sigma)_{v}$
corresponding to the same choice of smoothing. $S_{v}$ is the virtual closure
of $S$ since, $S=S_{v}\backslash\\{r\mid r\text{ is a virtual closure arc on
}E\\}$. By Corollary 3.1, $S$ and $S_{v}$ have the same number of components.
Thus
$\bigl{\langle}S^{\sigma}\bigr{\rangle}=\bigl{\langle}S_{v}\bigr{\rangle}$ for
all states $S$ of $L$. Hence,
$\bigl{\langle}L^{\sigma}\bigr{\rangle}=\bigl{\langle}(L_{\tau},\sigma)_{v}\bigr{\rangle}$.
Therefore, the Jones polynomial of $L$, according to Definition 4.3, is equal
to the Jones polynomial of its virtual closure, i.e.

$\displaystyle f_{L^{\sigma}}=(-A^{3})^{-\mathsf{Wr}(L)}\langle
L^{\sigma}\rangle=(-A^{3})^{-\mathsf{Wr}\left((L_{\tau},\sigma)_{v}\right)}\langle(L_{\tau},\sigma)_{v}\rangle=f_{(L_{\tau},\sigma)_{v}}.$
(7)

∎

######  Remark 4.3.

Theorem 4.2 allows to compute the Jones polynomial of virtual knots/links by
using linkoids and without using the closure arcs.

######  Example 4.1.

Let us consider the linkoid, $\displaystyle
L=\raisebox{-15.0pt}{\includegraphics[scale={0.5}]{fig/v-tref-linkoid-
bkt.png}}$, with strand permutation $\tau=(1\quad 2)(3\quad 4)$. The virtual
closure of $L$ with respect to $\sigma=(1\quad 4)(2\quad 3)$ corresponds to
the virtual trefoil knot, $\displaystyle
K=\raisebox{-12.0pt}{\includegraphics[scale={0.5}]{fig/v-tref-closed.png}}$.
Hence, the Jones polynomial of $L$ with respect to $\sigma$ is given as,
$f_{L^{\sigma}}=f_{K}=(-A^{3})^{-2}(A^{2}-A^{-4}+1)=A^{-4}-A^{-10}+A^{6}$.

###  4.3 The Arrow polynomial of Linkoids and Virtual Links

The construction of the arrow polynomial invariant begins with the oriented
state summation of the bracket polynomial where each local smoothing is either
an oriented smoothing or a disoriented smoothing, as shown in Equation 8, for
both positive and negative classical crossings in a diagram [22].

$\begin{split}\left\langle\raisebox{-7.0pt}{\includegraphics[width=19.5121pt]{fig/pos-
crs.png}}\right\rangle_{\mathcal{A}}=A\left\langle\raisebox{-7.0pt}{\includegraphics[width=19.5121pt]{fig/ori-
smooth.png}}\right\rangle_{\mathcal{A}}+A^{-1}\left\langle\raisebox{-7.0pt}{\includegraphics[width=17.34189pt]{fig/disori-
smooth.png}}\right\rangle_{\mathcal{A}},&\hskip
14.22636pt\left\langle\raisebox{-7.0pt}{\includegraphics[width=19.5121pt]{fig/neg-
crs.png}}\right\rangle_{\mathcal{A}}=A^{-1}\left\langle\raisebox{-7.0pt}{\includegraphics[width=19.5121pt]{fig/ori-
smooth.png}}\right\rangle_{\mathcal{A}}+A\left\langle\raisebox{-7.0pt}{\includegraphics[width=17.34189pt]{fig/disori-
smooth.png}}\right\rangle_{\mathcal{A}},\\\ \hskip
14.22636pt\bigl{\langle}\tilde{L}\cup\bigcirc\bigr{\rangle}_{\mathcal{A}}=d\bigl{\langle}\tilde{L}\bigr{\rangle}_{\mathcal{A}},&\hskip
14.22636ptd=-A^{2}-A^{-2},\end{split}$ (8)

For linkoid diagrams, any final state in the state sum expansion consists of
circular loops and/or segment cycles determined by the choice of closure
permutation. A loop or a segment cycle, which is obtained by disoriented
smoothings, also includes one or more paired cusps. Each cusp can be regarded
as an indication of one side of a given arc. The equivalence relation for
reduction of cusps is given in Figure 10. This means that consecutive
indications of both sides of an arc do not cancel, but consecutive indications
of one side do cancel. With this reduction, the resulting state sum is
invariant under the Reidemeister and virtual moves.

Figure 10: Equivalence moves on cusps in a state of the arrow polynomial.

The generalized arrow polynomial of a linkoid can be defined as follows:

######  Definition 4.5.

(Generalized arrow polynomial of a linkoid with respect to $\sigma$) Let $L$
be a linkoid diagram with $n$ components, $E$ be the set of its labelled
endpoints and $\sigma$ be a closure permutation. The generalized arrow
polynomial of $L$, with respect to $\sigma$, can be expressed as the following
state sum expression :

$\bigl{\langle}L^{\sigma}\bigr{\rangle}_{\mathcal{A}}\hskip 1.42271pt:=\hskip
1.42271pt\sum_{S}A^{\alpha(S)}d^{\operatorname{circ}(S)+|E/\langle\tau_{S},\sigma\rangle|-1}\mathcal{K}_{S}\quad,$
(9)

where $S$ is a state corresponding to a choice of smoothing (oriented or
disoriented) over all classical in $L$; $\alpha(S)$ is the algebraic sum of
the smoothing labels of $S$ and $\operatorname{circ}(S)$ is the total number
of circular components in $S$ and $|E/\langle\tau_{S},\sigma\rangle|$ is the
total number of segment cycles in $S$. Let $S_{i_{1}},S_{i_{2}},\ldots
S_{i_{m}}$, with $i_{1},i_{2}\ldots i_{m}$ pairs of surviving cusps
respectively, be all the state components in $S$ with surviving pairs of
cusps. Then $\langle S_{j}\rangle_{\mathcal{A}}=K_{j}$, for all
$j\in\\{i_{1},i_{2},\ldots i_{m}\\}$ and
$\mathcal{K}=\prod_{j=1}^{m}K_{i_{j}}$ (See Figure 11).

Figure 11: The arrow polynomial of a loop with $2i$ irreducible cusps is
denoted by the variable, $K_{i}$, where $i\in\mathbb{N}$.

######  Remark 4.4.

The generalized arrow polynomial of a linkoid is a polynomial in variables
$A,K_{i}$, where $i\in\mathbb{N}$ with integer coefficients.

######  Remark 4.5.

In [14], the arrow polynomial of knotoids was defined where a state in the
state sum expansion can have a cusped long segment. If the long segment has
$2i$ irreducible cusps, then the arrow polynomial of the long segment is
denoted by the variable $\Lambda_{i}$.

According to Theorem 4.1, the generalized arrow polynomial of a linkoid,
$L_{\tau}$, with respect to a closure permutation $\sigma$, can be defined as
the arrow polynomial of its virtual closure, $(L_{\tau},\sigma)_{v}$. This
defintion is equivalent to Defintion 4.5 due to the following therorem:

######  Theorem 4.3.

Let $L$ be a linkoid diagram with strand permutation, $\tau$ and $\sigma$ be a
closure permutation. The arrow polynomial of $L_{\tau}$, as in Definition 4.5,
coincides with the arrow polynomial of its virtual closure,
$(L_{\tau},\sigma)_{v}$.

###### Proof.

It follows similarly to Theorem 4.2. ∎

Example 4.2 shows the arrow polynomial state sum expansion for a particular
linkoid. One of the final states in this example corresponds to a loop with
cusps and virtual crossings. Via detour move, it can be seen that this state
is in fact a loop with irreducible cusps (See Figure 11).

######  Example 4.2.

The arrow polynomial of  is evaluated as follows :

$\begin{split}\left\langle\raisebox{-15.0pt}{\includegraphics[width=52.03227pt]{fig/CL-
main.png}}\right\rangle_{\mathcal{A}}&=\left\langle\raisebox{-18.0pt}{\includegraphics[width=43.36464pt]{fig/CL-
vc.png}}\right\rangle_{\mathcal{A}}=A\left\langle\raisebox{-15.0pt}{\includegraphics[width=43.36464pt]{fig/CL-
ori.png}}\right\rangle_{\mathcal{A}}+A^{-1}\left\langle\raisebox{-20.0pt}{\includegraphics[width=43.36464pt]{fig/CL-
disori.png}}\right\rangle_{\mathcal{A}}\\\
&=A+A^{-1}\left\langle\raisebox{-20.0pt}{\includegraphics[width=43.36464pt]{fig/CL-
detour.png}}\right\rangle_{\mathcal{A}}=A+A^{-1}K_{1}.\\\ \end{split}$ (10)

Example 4.3 illustrates the evaluatation the arrow polynomial of a 2 component
linkoid:

Figure 12: (Left) A linkoid $L$ with 2 components and strand permutation,
$\tau=(1\quad 4)(2\quad 3)$. (Right) The virtual closure of $L$, with respect
to the closure permutation, $\sigma=(1\quad 2)(3\quad 4)$, is equal to the
kishino knot, $K$.

######  Example 4.3.

Let us consider the linkoid, $L$, with strand permutation $\tau=(1\quad
4)(2\quad 3)$ and its virtual closure, with respect to $\sigma=(1\quad
2)(3\quad 4)$, $K$ as in Figure 12. Since, the Kishino knot has trivial Jones
polynomial, the Jones polynomial of $L$ with respect to $\sigma$ is also
trivial (despite $L$ being a non-trivial linkoid). However, the arrow
polynomial of $L$ is easily seen to be non-trivial, i.e. it matches the arrow
polynomial of $K$, namely, $A^{4}+1+A^{-4}-(A^{4}+2+A^{-4})K^{2}_{1}+2K_{2}.$
Thus, with respect to $\sigma$, $L$ is a linkoid that the Jones polynomial
cannot detect but the arrow polynomial is able to detect.

###  4.4 The Affine-index polynomial of Linkoids and Virtual Links

Given a labelled flat (without over/under information at crossings) virtual
knot diagram, two numbers, $W_{-}(c)$ and $W_{+}(c)$, can be defined at each
classical node $c$. If there is a labeled classical node with left incoming
arc $a$ and right incoming arc $b$, as in  , then the right outgoing arc is
labeled $a-1$ and the left outgoing arc is labeled $b+1$. Then
$W_{+}(c)=a-b-1$ and $W_{-}(c)=-a+b+1$ , respectively. Note that
$W_{-}(c)=-W_{+}(c)$ in all cases. Whereas, if there is a virtual node with
left incoming arc $a$ and right incoming arc $b$, then the right outgoing arc
is labeled $a$ and the left outgoing arc is labeled $b$ (i.e. No change at a
virtual crossing). The affine-index polynomial of a virtual knot was defined
in [26], in terms of such integer labelling on flat virtual knot diagrams, in
the following way:

######  Definition 4.6.

(Affine index polynomial of virtual knots) The Affine Index Polynomial of a
virtual knot diagram, $K$, is defined by the equation

$P_{K}:=\sum_{c}\operatorname{sgn}(c)\left(t^{W_{K}(c)}-1\right)=\sum_{c}\operatorname{sgn}(c)t^{W_{K}(c)}-\mathsf{wr}(K)\quad,$
(11)

where $c$ denotes a classical crossing in $K$, $\operatorname{sgn}(c)$ is the
sign of $c$, $\mathsf{wr}(K)$ is the writhe of $K$ and
$W_{K}(c)=W_{\operatorname{sgn}(c)}(c)$, obtained from labelling of arcs in
the flat counterpart of $K$.

According to Theorem 4.1, the affine-index polynomial of a linkoid,
$L_{\tau}$, with respect to a closure permutation $\sigma$, can be defined as
the affine-index polynomial of its virtual closure, $(L_{\tau},\sigma)_{v}$
provided $(L_{\tau},\sigma)_{v}$ consists of only one component.

######  Definition 4.7.

(Affine-index polynomial of a linkoid with respect to $\sigma$) Let $L$ be a
linkoid diagram with $n$ components, $E$ be the set of its labelled endpoints
and $\sigma$ be a closure permutation such that $(L_{\tau},\sigma)_{v}$
consists of only one component. The affine-index polynomial of $L$, with
respect to $\sigma$, is defined as:

$P_{L^{\sigma}}:=P_{(L_{\tau},\sigma)_{v}}\quad,$ (12)

where $P_{(L_{\tau},\sigma)_{v}}$ is the affine -index polynomial of the
virtual knot, $(L_{\tau},\sigma)_{v}$.

Example 4.4 illustrates the evaluatation the affine-index polynomial of a 2
component linkoid:

######  Example 4.4.

Consider the linkoid, $\displaystyle
L=\raisebox{-15.0pt}{\includegraphics[scale={0.4}]{fig/affine-linkoid.png}}$,
with 2 components. The virtual closure of $L$, with respect to $\sigma=(1\quad
2)(3\quad 4)$, has only 1 component whose flat diagram is given as,
$\displaystyle FV=\raisebox{-15.0pt}{\includegraphics[scale={0.4}]{fig/affine-
VK.png}}$. The labels $A,B$ and $C$ correspond to classical crossings, which
all have positive signs. Thus, $W_{+}(A)=-2,W_{+}(B)=2$ and $W_{+}(C)=0$ are
respective the weights associated with the classical crossings. The virtual
knot, $(L_{\tau},\sigma)_{v}$ is one with unit Jones polynomial. However, it
has a non-trivial affine-index polynomial, namely,
$P_{(L_{\tau},\sigma)_{v}}=t^{-2}+t^{2}-2$. In other words, the linkoid $L$
has trivial Jones polynomial but non-trivial affine-index polynomial, with
respect to $\sigma=(1\quad 2)(3\quad 4)$.

###  4.5 The Odd Writhe of Linkoids and Virtual Knots

The odd writhe of a virtual knot, which was shown to be an invariant of
virtual knots in [27], is defined as follows:

######  Definition 4.8.

(Odd crossing and Odd writhe of a virtual knot) A crossing of a virtual knot
is called odd if its crossing labels in the Gauss diagram have an odd number
of crossing labels between them. The odd writhe of the virtual knot is defined
as the sum of the signs of the odd crossings.

Note that in classical knots, the odd writhe is always zero, since all
crossings of classical knots are even. As an example, the virtual closure of
the linkoid in Figure 13 has odd writhe equal to $-1$, which proves that this
knot is not equivalent to any classical knot.

Figure 13: A linkoid diagram with strand permutation $\tau=(1\quad 2)(3\quad
4)$, its virtual closure with respect to $\sigma=(1\quad 3)(2\quad 4)$ and its
Gauss diagram. The crossings labelled, $4,5,6$ and $7$ are odd and the odd-
writhe of the virtual knot diagram is $-1$.

Due to Theorem 4.1, the odd writhe of linkoids is well-defined with respect to
closure permutations which gives rise to a virtual knot (one component):

######  Definition 4.9.

(Odd writhe of a linkoid with respect to $\sigma$) Let $L$ be a linkoid
diagram and $\sigma$ be a closure permutation such that, $(L,\sigma)_{v}$ has
one component. The odd writhe, $\mathsf{owr}$ of $L$, with respect to
$\sigma$, is defined as the sum of the signs of the odd crossings in
$(L,\sigma)_{v}$.

######  Remark 4.6.

For linkoids, the odd-writhe is an integer valued invariant.

######  Remark 4.7.

In the context of virtual links (2 or more components), the notion of parity
does not extend naturally. Depending upon how the link components are
examined, there is an ambiguity regarding whether a crossing is odd or even.
This ambiguity can be remedied by defining even and odd for crossings in
individual components while labeling crossings shared by 2 components as link
crossings [28].

##  5 The Virtual Spectrum of a Linkoid

This section introduces the virtual spectrum of linkoids which leads to new
invariants of linkoids that are not those of any given virtual closure. For
any linkoid with fixed strand permutation, the virtual spectrum is the set of
virtual links resulting from virtual closure over all possible closure
permutations. More formally, it can be defined as follows:

######  Definition 5.1.

(Virtual spectrum of a linkoid) The virtual spectrum of a linkoid, $L$, with
$n$ components and strand permutation, $\tau$, is defined to be the set,
$\operatorname{spec}(L):=\displaystyle\\{\varphi_{n}(L,\tau,\sigma):\sigma\in
H_{n}\\}$, where $\varphi_{n}$ is the virtual closure map defined on linkoids
with $n$ components, $\displaystyle E=\\{1,2,\ldots,2n\\}$ and $\displaystyle
H_{n}=\\{\sigma\in S_{2n}:\sigma^{2}(i)=i,\sigma(i)\neq i,\quad\forall i\in
E\\}$. In other words, the virtual spectrum of $L$ is given by
$\displaystyle\operatorname{Im}({\varphi_{n}}{\big{|}}_{L,\tau})$.

######  Theorem 5.1.

For a linkoid, $L$, its virtual spectrum, $\operatorname{spec}(L)$ is an
invariant of $L$.

###### Proof.

By Definition 5.1, each element in $\operatorname{spec}(L)$ is of the form,
$(L,\sigma)_{v}$, where $\sigma$ is a closure permutation. For any $\sigma$,
the virtual closure, $(L,\sigma)_{v}$, is a linkoid invariant by Remark 3.2
and Proposition 3.3. Hence, $\operatorname{spec}(L)$ is an invariant of $L$. ∎

######  Corollary 5.1.

If two linkoid diagrams, $L_{1}$ and $L_{2}$, are equivalent up to
Reidemeister moves, then
$\operatorname{spec}(L_{1})=\operatorname{spec}(L_{2})$.

###### Proof.

It follows from Theorem 5.1 ∎

######  Corollary 5.2.

Let $L_{1}$ and $L_{2}$ be two linkoid diagrams. If
$\operatorname{spec}(L_{1})\neq\operatorname{spec}(L_{2})$, then $L_{1}$ and
$L_{2}$ are distinct.

###### Proof.

It follows from Theorem 5.1. ∎

######  Remark 5.1.

Two non-equivalent linkoids will have the same virtual spectrum only if all
the closure permutations give the same virtual knots/links.

######  Example 5.1.

Note that,  and  are two non-equivalent linkoids with same number of
components and same virtual spectrum.

######  Remark 5.2.

Property ii of the virtual closure map implies that the virtual spectrum of a
trivial linkoid with $n$-components has size $n$.

###  5.1 New Invariants of Linkoids from the virtual spectrum

New invariants of linkoids can be defined, independent of any particular
closure permutation, by means of the virtual spectrum of linkoids.

######  Definition 5.2.

(Average spectral $\mathcal{F}$-invariant of a linkoid) Let $\mathcal{F}$ be
an invariant of virtual links, $L$ be a linkoid and
$\displaystyle\mathcal{F}(\operatorname{spec}(L))$ be the collection of values
of $\mathcal{F}$ over all virtual links in $\operatorname{spec}(L)$. The
average spectral $\mathcal{F}$-invariant of $L$ is defined as the average of
$\displaystyle\mathcal{F}(\operatorname{spec}(L))$, namely :

$\displaystyle\operatorname{avg}_{\mathcal{F}}(L):=\displaystyle\overline{\mathcal{F}(\operatorname{spec}(L))}=\frac{1}{|\operatorname{spec}(L)|}\sum_{\ell\in\operatorname{spec}(L)}\mathcal{F}(\ell).$

The invariance of the average spectral $\mathcal{F}$-invariant of a linkoid is
given by the following theorem:

######  Theorem 5.2.

Let $L$ be a linkoid, $\operatorname{spec}(L)$ be its virtual spectrum and let
$\mathcal{F}$ be an invariant of virtual links. Let
$\displaystyle\mathcal{F}(\operatorname{spec}(L))$ be the collection of values
of $\mathcal{F}$ over all virtual links in $\operatorname{spec}(L)$. Then, the
set, $\displaystyle\mathcal{F}(\operatorname{spec}(L))$ and the measure,
$\operatorname{avg}_{\mathcal{F}}(L)$, are invariants of the linkoid, $L$.

###### Proof.

Since, $\mathcal{F}$ is an invariant of virtual links, there exits an
invariant of linkoids, $\displaystyle{\mathcal{F}_{\sigma}}$, which is defined
as $\mathcal{F}$ acting on the closure of the linkoid with respect to a
closure permutation, $\sigma$ ( by Theorem 4.1). Therefore, the elements of
the set, $\displaystyle\mathcal{F}(\operatorname{spec}(L))$, are of the form,
$\displaystyle{\mathcal{F}_{\sigma}}(L)$, where $\sigma$ is a closure
permutation. Each $\mathcal{F}_{\sigma}$ is an invariant of $L$, hence their
collection, $\displaystyle\mathcal{F}(\operatorname{spec}(L))$ and the
measure, $\operatorname{avg}_{\mathcal{F}}(L)$, are also invariants of $L$. ∎

In this way, we can define the new invariants of linkoids such as, average
spectral height, average spectral genus, average spectral Jones polynomial and
average spectral arrow polynomial. The average spectral height and genus of
linkoids are rational numbers whereas, the average spectral Jones polynomial
and arrow polynomial of linkoids are polynomials with rational coefficients.

######  Example 5.2.

For the linkoid,  with $\tau=(1\quad 2)(3\quad 4)$, the virtual spectrum is
given by accounting for all $\sigma\in\\{(1\quad 2)(3\quad 4),(1\quad
3)(2\quad 4),(1\quad 4)(2\quad 3)\\}$, i.e.
$\operatorname{spec}\left(\raisebox{-7.0pt}{\includegraphics[scale={0.05}]{fig/hopf-
open.png}}\right)=\left\\{\raisebox{-7.0pt}{
\includegraphics[scale={0.07}]{fig/hc.png},
\includegraphics[scale={0.07}]{fig/hairpin.png},
\includegraphics[scale={0.07}]{fig/vir-tref1.png}}\right\\}$. Let $\mathsf{h}$
denote the height invariant of linkoids. Then,
$\mathsf{h}\left(\raisebox{-7.0pt}{
\includegraphics[scale={0.07}]{fig/hc.png}}\right)=\mathsf{h}\left(\raisebox{-7.0pt}{\includegraphics[scale={0.07}]{fig/hairpin.png}}\right)=0$
and $\mathsf{h}\left(\raisebox{-7.0pt}{
\includegraphics[scale={0.07}]{fig/vir-tref1.png}}\right)=1$. Hence,
$\mathsf{h}\left(\operatorname{spec}\left(\raisebox{-7.0pt}{\includegraphics[scale={0.05}]{fig/hopf-
open.png}}\right)\right)=\\{0,0,1\\}$ and
$\operatorname{avg}_{\mathsf{h}}\left(\raisebox{-7.0pt}{\includegraphics[scale={0.05}]{fig/hopf-
open.png}}\right)=1/3$.

######  Corollary 5.3.

If two linkoid diagrams, $L_{1}$ and $L_{2}$, are equivalent up to
Reidemeister moves, then
$\mathcal{F}(\operatorname{spec}(L_{1}))=\mathcal{F}(\operatorname{spec}(L_{2}))$
and
$\operatorname{avg}_{\mathcal{F}}(L_{1})=\operatorname{avg}_{\mathcal{F}}(L_{2})$,
where $\mathcal{F}$ is an invariant of virtual links.

###### Proof.

It follows from Corollary 5.1 and Theorem 5.2. ∎

######  Corollary 5.4.

Let $L_{1}$ and $L_{2}$ be two arbitrary linkoid diagrams and $\mathcal{F}$ be
an invariant of virtual links. If
$\mathcal{F}(\operatorname{spec}(L_{1}))\neq\mathcal{F}(\operatorname{spec}(L_{2}))$,
then $L_{1}$ and $L_{2}$ are distinct. Similarly, if
$\operatorname{avg}_{\mathcal{F}}(L_{1})\neq\operatorname{avg}_{\mathcal{F}}(L_{2})$,
then $L_{1}$ and $L_{2}$ are distinct.

###### Proof.

It follows from Theorem 5.2. ∎

######  Corollary 5.5.

The $\mathcal{F}$-invariant of a linkoid over its virtual spectrum is stronger
than the invariant $\mathcal{F}$ applied to any individual closure unless the
invariant is a complete classifier of virtual knots/links. Similarly, the
average $\mathcal{F}$-invariant, $\operatorname{avg}_{\mathcal{F}}$, is a
stronger classifier of a linkoid than the invariant $\mathcal{F}$ applied to
any individual closure, unless the invariant is a complete classifier of
virtual knots/links.

###### Proof.

It follows from Theorem 5.2. ∎

Through the following example, it is shown that evaluating an invariant over
the virtual spectrum can better tell apart linkoids compared to evaluating the
invariant for any particular closure permutation.

######  Example 5.3.

Let us consider the linkoids, $\displaystyle
L_{1}=\raisebox{-10.0pt}{\includegraphics[scale={0.2}]{fig/kishino-
linkoid.png}}$ and $\displaystyle
L_{2}=\raisebox{-10.0pt}{\includegraphics[scale={0.4}]{fig/linkoid-X.png}}$,
each with strand permutation $\tau=(1\quad 4)(2\quad 3)$. Note that the Jones
polynomial of both $L_{1}$ and $L_{2}$, with respect to closure permutation,
$\sigma=(1\quad 2)(3\quad 4)$, is equal to $1$. However, the spectrum of the
Jones polynomial of $L_{1}$ over all closure permutations is $\\{1,\quad
2-2A^{-2}-A^{2}-A^{-4},\quad-2A^{4}+2A^{-2}+2-2A^{2}-2A^{4}\\}$. Whereas, the
spectrum of the Jones polynomial of $L_{2}$ over all closure permutations is
$\\{1,\quad-A^{4}-A^{2},\quad 1\\}$. Since the spectrum of the Jones
polynomial for the two linkoids are different, hence $L_{1}$ and $L_{2}$ are
non-equivalent as linkoids.

In addition to the average spectral $\mathcal{F}$-invariant of a linkoid, the
minimal spectral $\mathcal{F}$-invariant of a linkoid can be defined as
follows for any $\mathcal{F}$-invariant of virtual links, which takes values
in $\mathbb{R}$.

######  Definition 5.3.

(Minimal spectral $\mathcal{F}$-invariant of a linkoid) Let $\mathcal{F}$ be
an invariant of virtual links which takes values in $\mathbb{R}$ (such as
height, genus and odd writhe). The minimal spectral $\mathcal{F}$-invariant of
a linkoid, $L$, is defined as the minimum of the values of the
$\mathcal{F}$-invariant over all elements in the virtual spectrum,
$\operatorname{spec}(L)$:

$\displaystyle\operatorname{min}_{\mathcal{F}}(L):=\operatorname{min}\left\\{\mathcal{F}(\ell):\ell\in\operatorname{spec}(L)\right\\}.$

######  Remark 5.3.

The minimal spectral height and genus of a linkoid can be defined due to
Definition 5.3 and they take values in $\mathbb{Z}$. Whereas, the average
spectral height and genus of a linkoid takes values in $\mathbb{Q}$.

##  6 Measures of entanglement of Open Curves in 3-space via Virtual Closure

In this section, it is shown how topological invariants from virtual knot
theory can be extended to define measures of entanglement of open curves in
3-space.

For a collection of open or closed curves in 3-space in general position, any
(regular) projection of these curves gives rise to a linkoid diagram. In fact,
any such projection will be generic with probability one. Based on the
directions of projection, given as unit vectors in $S^{2}$, it is possible to
get non-equivalent linkoid diagrams for a collection of open curves in
3-space. To fully capture the entanglement complexity of a collection of open
curves in 3-space, it is necessary to account for all possible directions of
projection [6, 5, 4, 29].

In the following discussion, let $\mathcal{L}$ be a collection of $n$ open
curves in 3-space, with endpoints denoted by labels from the set
$E=\\{1,2,\cdots,2n\\}$, without repetition. A strand permutation and a
closure permutation of $\mathcal{L}$ can be defined in the same way as
Definitions 3.2 and 3.3, namely:

######  Definition 6.1.

(Strand permutation of collections of open curves in 3-space) The strand
permutation of $\mathcal{L}$ is defined to be the element, $\tau\in S_{2n}$,
such that, for any $i\in E$, $i$ and $\tau(i)$ are labels for the two
endpoints of the same component of $\mathcal{L}$.

######  Definition 6.2.

(Closure permutation of collections of open curves in 3-space) A closure
permutation of $\mathcal{L}_{\tau}$, where $\tau$ denotes a strand permutation
of $\mathcal{L}$, is any element, $\sigma\in S_{2n}$, such that,
$\sigma^{2}=\operatorname{id}$ and $\sigma(i)\neq i$, for all $i\in E$.

Notice that, for every $\vec{\xi}\in S^{2}$, the projection of $\mathcal{L}$
to the plane with normal vector, $\vec{\xi}$ is a linkoid diagram,
$\mathcal{L}_{\vec{\xi}}$. A choice of strand permutation and closure
permutation of a collection of open curves, $\mathcal{L}$, canonically induces
a strand permutation and closure permutation on a projection,
$\mathcal{L}_{\vec{\xi}}$, for all $\vec{\xi}\in S^{2}$. Therefore,
$\displaystyle\left(\mathcal{L}_{\vec{\xi}}\right)_{\tau}$ and
$\displaystyle\left(\left(\mathcal{L}_{\vec{\xi}}\right)_{\tau},\sigma\right)_{v}$
are well-defined for all $\vec{\xi}\in S^{2}$, where $\tau$ and $\sigma$ are
the given strand permutation and closure permutation of $\mathcal{L}$. The
notion of virtual closure can be extended to open curves in 3-space by
introducing virtual closure arcs in 3-space according to some closure
permutation (See Figure 14), as defined before.

Figure 14: Projections of the virtual closure of open curves in 3-space giving
rise to virtual knot diagrams.

######  Definition 6.3.

(Virtual arc in 3-space) A virtual arc in 3-space is an arc in 3-space such
that its projection in any direction involves only virtual crossings.

######  Definition 6.4.

(Virtual closure of collections of open curves in 3-space with respect to
$\sigma$) Let $\mathcal{L}$ be a collection of open curves in 3-space with
strand permutation, $\tau$. The virtual closure of $\mathcal{L}_{\tau}$, with
respect to $\sigma$, is denoted by
$\displaystyle\left(\mathcal{L}_{\tau},\sigma\right)_{v}$ and it is defined to
be the knot/link in 3-space, with virtual closure arcs between $i$ and
$\sigma(i)$, for all $i\in E$.

######  Corollary 6.1.

Let $\mathcal{L}$ denote a collection of open curves in 3-space and let $\tau$
and $\sigma$ denote its strand permutation and closure permutation,
respectively. Let $\vec{\xi}\in S^{2}$ and let
$\displaystyle\left(\left(\mathcal{L}_{\tau},\sigma\right)_{v}\right)_{\vec{\xi}}$
be the projection, with respect to $\vec{\xi}$, of the virtual closure of
$\mathcal{L}_{\tau}$, with respect to $\sigma$ and
$\displaystyle\left(\left(\mathcal{L}_{\vec{\xi}}\right)_{\tau},\sigma\right)_{v}$
be the virtual closure of the linkoid diagram,
$\displaystyle\left(\mathcal{L}_{\vec{\xi}}\right)_{\tau}$, with respect to
$\sigma$. Then
$\displaystyle\left(\left({\mathcal{L}_{\tau}},\sigma\right)_{v}\right)_{\vec{\xi}}=\displaystyle\left(\left(\mathcal{L}_{\vec{\xi}}\right)_{\tau},\sigma\right)_{v}$.

###### Proof.

By Definition 6.4, all the virtual crossings in
$\displaystyle\left(\left({\mathcal{L}_{\tau}},\sigma\right)_{v}\right)_{\vec{\xi}}$
are contributed by the projections of the virtual closure arcs in
$\displaystyle\left({\mathcal{L}_{\tau}},\sigma\right)_{v}$. The linkoid
diagram, $\displaystyle\left(\mathcal{L}_{\vec{\xi}}\right)_{\tau}$, is
obtained if the arcs between $i$ and $\sigma(i)$ are removed from the diagram,
$\left(\left({\mathcal{L}_{\tau}},\sigma\right)_{v}\right)_{\vec{\xi}}$, for
all $i\in\\{1,2,\ldots,2n\\}$. Hence,
$\displaystyle\left(\left({\mathcal{L}_{\tau}},\sigma\right)_{v}\right)_{\vec{\xi}}=\displaystyle\left(\left(\mathcal{L}_{\vec{\xi}}\right)_{\tau},\sigma\right)_{v}$.
∎

The following corollary asserts that the virtual closure of collection of open
curves in 3-space is independent of the embedding of the virtual arcs in
3-space.

######  Corollary 6.2.

Let $\mathcal{L}$ denote a collection of open curves in 3-space and let $\tau$
and $\sigma$ denote its strand permutation and closure permutation,
respectively. Let $X$ and $Y$ denote two representations of
$\displaystyle\left(\mathcal{L}_{\tau},\sigma\right)_{v}$, corresponding to
different embeddings of the virtual arcs. Then, $X_{\vec{\xi}}=Y_{\vec{\xi}}$
for all $\vec{\xi}\in S^{2}$.

###### Proof.

It follows from Corollary 6.1. ∎

A projection of a collection of open curves in 3-space can give only finitely
many knotoids/linkoids [29, 6]. Similarly, for any closure permutation, the
virtual closure of a collection of open curves in 3-space can be associated
with a finite family of virtual knot diagrams (which arise from projections of
the virtual closure).

######  Corollary 6.3.

Let $\mathcal{L}$ be a collection of open curves in 3-space. Given a closure
permutation, $\sigma$, the number of distinct virtual knots associated with
the virtual closure of $\mathcal{L}$ is less than or equal to the number of
distinct knotoids/linkoids associated with $\mathcal{L}$.

###### Proof.

A projection of $\mathcal{L}$ can give only finitely many knotoids/linkoids
[29, 6]. Hence, a projection of the virtual closure of $\mathcal{L}$, with
respect to $\sigma$, can give only finitely many virtual knots/links. The
claim follows from the fact that the virtual closure map on linkoids is not
injective. (See Theorem 3.1). ∎

######  Remark 6.1.

By Definition 6.4, a new notion of virtual links in 3-space arises i.e. the
links in 3-space containing one or more virtual arcs. Let us denote virtual
links in 3-space by $\mathcal{V}^{3D}$. Let for every $n\in\mathbb{N}$,
$\mathcal{A}_{n}^{3D}$ be the set of all collections of open curves in 3-space
with $n$ components and $H_{n}=\\{\sigma\in
S_{2n}:\sigma^{2}(i)=i,\sigma(i)\neq i,\quad\forall i\in E\\}$, where
$E=\\{1,2,\ldots,2n\\}$. Similar to Definition 3.8, we can define a closure
map relating open curves in 3-space to virtual links in 3-space as follows :

$\begin{split}\phi&:\quad\\{(\mathcal{L},\tau,\sigma)\mid(\mathcal{L},\tau,\sigma)\in\mathcal{A}_{n}^{3D}\times
H_{n}\times H_{n}\text{ for some
}n\in\mathbb{N}\\}\longrightarrow\mathcal{V}^{3D}\\\ &\hskip
142.26378pt(\mathcal{L},\tau,\sigma)\mapsto(\mathcal{L}_{\tau},\sigma)_{v}\hskip
14.22636pt,\\\ \end{split}$

where $\tau,\sigma\in H_{n}$, $\mathcal{L}\in\mathcal{A}_{n}^{3D}$ and
$(\mathcal{L}_{\tau},\sigma)_{v}$ is the virtual closure of
$\mathcal{L}_{\tau}$ with respect to $\sigma$. The map $\phi$ has properties
analogous to properties i-iv of the virtual closure map $\varphi$ of linkoids
(See Defintion 3.8).

######  Remark 6.2.

Through virtual closure, open curves in 3-space can be associated to a
collection of embeddings of links in thickened surfaces of arbitrary genus.

A collection of new measures of entanglement of collections of open curves in
3-space can be defined by using invariants of virtual knots as follows :

######  Definition 6.5.

(New measures of entanglement of collections of open curves in 3-space) Let
$\mathcal{L}$ be a collection of open curves in 3-space and let $\mathcal{F}$
be any invariant of virtual links. Given a closure permutation $\sigma$, a
measure of entanglement of $\mathcal{L}_{\tau}$ is,

$\displaystyle\mathcal{F}_{\sigma}({\mathcal{L}_{\tau}})\hskip
1.42271pt:=\hskip 1.42271pt\frac{1}{4\pi}\int_{\vec{\xi}\in
S^{2}}\mathcal{F}\left(\left(\mathcal{L}_{\tau},\sigma\right)_{\vec{\xi}}\right)dS\hskip
1.42271pt.$ (13)

where $\left(\mathcal{L}_{\tau},\sigma\right)_{\vec{\xi}}$ denotes the
projection along $\vec{\xi}$ of the virtual closure of $\mathcal{L}_{\tau}$,
with respect to $\sigma$.

Note that the integral in Equation 13 is taken over all vectors $\vec{\xi}\in
S^{2}$ except a set of measure zero (corresponding to the irregular
projections). For collections of open curves in 3-space,
$\mathcal{F}_{\sigma}$ has the following properties :

  1. 1.

$\mathcal{F}_{\sigma}$ does not depend on any particular projection of the
collection of open or closed curves.

  2. 2.

For a collection of open curves, $\mathcal{F}_{\sigma}$ is not an invariant of
a corresponding/approximating link, linkoid or virtual link.

  3. 3.

$\mathcal{F}_{\sigma}$ is not a topological invariant, but it is a continuous
function of the curve coordinates (see Corollary 6.4).

  4. 4.

For a collection of closed curves in 3-space (a link), $\mathcal{F}_{\sigma}$
coincides with a link invariant and it can be computed from a single
projection, i.e.
$\mathcal{F}_{\sigma}\left(\mathcal{L}_{\tau}\right)=\mathcal{F}_{\sigma}\left(\left(\mathcal{L}_{\vec{\xi}}\right)_{\tau}\right)=\mathcal{F}\left(\left(\mathcal{L}_{\tau},\sigma\right)_{\vec{\xi}}\right)$,
where $\vec{\xi}\in S^{2}$ is any projection vector.

  5. 5.

As the endpoints of a collection of open curves in 3-space tend to coincide,
according to $\sigma$, the value of $\mathcal{F}_{\sigma}$ tends to the value
of $\mathcal{F}$ on the corresponding classical link in 3-space (see Corollary
6.4).

######  Corollary 6.4.

Let $\mathcal{L}$ denote a collection of open curves in 3-space. Then,
$\mathcal{F}_{\sigma}$ is a continuous function of the curve coordinates of
$\mathcal{L}_{\tau}$. As the endpoints of $\mathcal{L}_{\tau}$ tend to
coincide according to $\sigma$, to form a closed link in 3-space,
$\mathcal{F}_{\sigma}$ tends to the value of $\mathcal{F}$ on the
corresponding classical link in 3-space.

###### Proof.

Let us approximate $\mathcal{L}$ by a set of polygonal curves of $n$ edges
each, namely $\mathcal{L}^{(n)}$. Then,

$\displaystyle\mathcal{F}_{\sigma}\left({\mathcal{L}^{(n)}}_{\tau}\right)=\sum_{i=1}^{k}p_{i}\mathcal{F}_{\sigma}\left({\mathsf{L}^{(n)}_{i}}_{\tau}\right),$
(14)

where ${\mathsf{L}^{(n)}_{i}}_{\tau}$ for $i=1,\dotsc,k$ are the possible
linkoids that can occur in all projections of ${\mathcal{L}^{(n)}}_{\tau}$ and
$p_{i}$, the corresponding geometric probabilities. The geometric probability
$p_{i}$ can be expressed as, $\displaystyle p_{i}=\frac{2A_{0}}{4\pi}$, where
$A_{0}$ is the area on the sphere corresponding to unit vectors such that the
projection of ${\mathcal{L}^{(n)}}_{\tau}$ along such vectors results in the
linkoid ${\mathsf{L}^{(n)}_{i}}_{\tau}$. $A_{0}$ is the area of the quadrangle
bounded by great circles defined by the edges and vertices of the polygonal
curves in ${\mathcal{L}^{(n)}}_{\tau}$, which is a continuous function of the
coordinates of ${\mathcal{L}^{(n)}}_{\tau}$ (see proof of Lemma 3.1 in [4]).
The result follows as $n$ goes to infinity.

As the endpoints tend to coincide, the geometric probability that a projection
gives a pure linkoid (or a pure virtual link) goes to $0$, while the geometric
probability that it gives the link-type linkoid goes to $1$. Thus, the value
of $\mathcal{F}_{\sigma}$ on $\mathcal{L}_{\tau}$ tends to the value of
$\mathcal{F}$ on the corresponding classical link in 3-space. ∎

For example, the height, genus and odd-writhe for a collection of open curves
in 3-space can be defined as follows :

######  Definition 6.6.

(Height of a collection of open curves in 3-space with respect to $\sigma$)
Let $\mathcal{L}$ denote a collection of open curves in 3-space. Given a
closure permutation $\sigma$, the height of a collection of open curves in
3-space, $\mathcal{L}_{\tau}$, is defined as,

$\displaystyle h_{\sigma}({\mathcal{L}_{\tau}})\hskip 1.42271pt:=\hskip
1.42271pt\frac{1}{4\pi}\int_{\vec{\xi}\in
S^{2}}h\left(\left(\mathcal{L}_{\tau},\sigma\right)_{\vec{\xi}}\right)dS\hskip
1.42271pt.$ (15)

where $\left(\mathcal{L}_{\tau},\sigma\right)_{\vec{\xi}}$ denotes a
projection of the virtual closure of $\mathcal{L}_{\tau}$ with respect to
$\sigma$.

######  Definition 6.7.

(Genus of a collection of open curves in 3-space with respect to $\sigma$ )
Let $\mathcal{L}$ denote a collection of open curves in 3-space. Given a
closure permutation $\sigma$, the genus of a collection of open curves in
3-space, $\mathcal{L}_{\tau}$, is defined as,

$\displaystyle g_{\sigma}({\mathcal{L}_{\tau}})\hskip 1.42271pt:=\hskip
1.42271pt\frac{1}{4\pi}\int_{\vec{\xi}\in
S^{2}}g\left(\left(\mathcal{L}_{\tau},\sigma\right)_{\vec{\xi}}\right)dS\hskip
1.42271pt.$ (16)

where $\left(\mathcal{L}_{\tau},\sigma\right)_{\vec{\xi}}$ denotes a
projection of the virtual closure of $\mathcal{L}_{\tau}$ with respect to
$\sigma$.

######  Definition 6.8.

(Odd writhe of a collection of open curves in 3-space with respect to
$\sigma$) Let $\mathcal{L}$ denote a collection of open curves in 3-space and
let $\sigma$ be a closure permutation such that, $(\mathcal{L},\sigma)_{v}$
has only one component. Then, the odd-writhe of a collection of open curves in
3-space, is defined as,

$\displaystyle\mathsf{owr}_{\sigma}({\mathcal{L}_{\tau}})\hskip
1.42271pt:=\hskip 1.42271pt\frac{1}{4\pi}\int_{\vec{\xi}\in
S^{2}}\mathsf{owr}\left(\left(\mathcal{L}_{\tau},\sigma\right)_{\vec{\xi}}\right)dS\hskip
1.42271pt.$ (17)

where $\left(\mathcal{L}_{\tau},\sigma\right)_{\vec{\xi}}$ denotes a
projection of the virtual closure of $\mathcal{L}_{\tau}$ with respect to
$\sigma$.

Similarly, we can define the Jones polynomial and the arrow polynomial of
collections of open curves in 3-space with respect to a given closure
permutation. The affine-index polynomial of collections of open curves in
3-space is also well-defined provided the closure permutation in concern gives
rise to a virtual link in 3-space with only 1 component.

Notice that for open curves in 3-space, the height, genus and odd-writhe are
real numbers that are continuous functions of the curve coordinates and they
tend to the integer topological invariant as the endpoints coincide. The Jones
polynomial, arrow polynomial and affine-index polynomial for open curves in
3-space have real valued coefficients which are continuous functions of the
curves co-ordinates. As the endpoints of the curves tend to coincide, these
polynomials tend to the corresponding polynomial-type invariants of the
resultant closed knot/link in 3-space.

######  Remark 6.3.

The virtual closure of open curves in 3-space can be useful in the context of
physical systems that employ periodic boundary conditions [30, 1, 31, 32]. The
generating cell, $C$, of a PBC system is in fact an $n$-tangle, $T$, possibly
involving endpoints inside the tangle. Here $n\in\mathbb{N}$ determines the
number of arcs inside $C$ and the periodic conditions canonically determine a
strand permutation, $\tau_{pbc}$, for $T$. The virtual closure of $T$, denoted
$(T_{\tau_{pbc}},\tau_{pbc})$, can be formed by introducing closure arcs to
the endpoints in $T$, lying on the boundary surface of the $C$, with respect
to $\tau_{pbc}$. In any projection of $(T_{\tau_{pbc}},\tau_{pbc})$, the arcs
inside $C$ contributes classical crossings to the diagram whereas the arcs
(virtual closure arcs) lying outside $C$ contributes virtual crossings to the
diagram (See Figures 15 and 16). Thus, for any projection of a PBC box, we can
assign a virtual link. The virtual link however, depends on the direction of
projection of the box.

Figure 15: Virtual closure of the arcs in a base cell of a PBC system in 3D
and one of its projections (virtual knot diagram).

By Theorem 6.5, novel measures of entanglement for systems employing PBC can
be defined in terms of the virtual closure (respecting PBC) of the arcs within
a base cell.

Figure 16: The virtual closure of the generating cell of two different systems
employing 2 periodic boundary conditions.

###  6.1 The Virtual Spectrum of collections of Open Curves in 3-space

In this section, the virtual spectrum and the weighted virtual spectrum of
collections of open curves in 3-space is discussed to define a new measure of
entanglement of open curves in 3-space that is independent of any virtual
closure.

######  Definition 6.9.

(Virtual Spectrum of open curves in 3-space) Let $\mathcal{L}$ be a collection
of open curves in 3-space with $n$ components and strand permutation, $\tau$
on the set of endpoints, $E$. The virtual spectrum of $\mathcal{L}_{\tau}$ is
defined to be the set,
$\displaystyle\operatorname{spec}(\mathcal{L}_{\tau}):=\displaystyle\left\\{(\mathcal{L}_{\tau},\sigma)_{v}:\sigma\in
H_{n}\right\\}$, where $H_{n}$ is the set of all closure permutations on $E$.

######  Definition 6.10.

(Weighted Virtual Spectrum of open curves in 3-space) Let $\mathcal{L}$ be a
collection of open curves in 3-space with $n$ components and strand
permutation, $\tau$ on the set of endpoints, $E$. The weighted virtual
spectrum of $\mathcal{L}_{\tau}$ is defined to be the set,

$\displaystyle\operatorname{spec}(\mathcal{L}_{\tau}):=\displaystyle\left\\{\left(w_{\sigma},(\mathcal{L}_{\tau},\sigma)_{v}\right):\sigma\in
H_{n},\quad w_{\sigma}=\frac{1}{2n}\sum_{i\in
E}\operatorname{d}(i,\sigma(i))\right\\}$, where $H_{n}$ is the set of all
closure permutations on $E$ and $\operatorname{d}(i,\sigma(i))$ denotes the
Euclidean distance between the points $i$ and $\sigma(i)$ for all $i\in E$.

Using the weighted virtual spectrum, we define a new measure of entanglement
for collections of open curves in 3-space:

######  Definition 6.11.

(Spectral measure of open curves in 3-space) Let $\mathcal{L}$ be a collection
of open curves in 3-space with $n$ components and strand permutation, $\tau$
on the set of endpoints, $E$. Let $\mathcal{F}$ be an invariant of virtual
knots/links. The $\mathcal{F}$-spectral measure of $\mathcal{L}_{\tau}$ is
defined as:

$\displaystyle\mathcal{F}(\mathcal{L}_{\tau}):=\sum_{\sigma\in
H_{n}}\frac{w_{min}}{w_{\sigma}}\mathcal{F}_{\sigma}\left(({\mathcal{L}}_{\tau},\sigma)_{v}\right),$

where $\displaystyle w_{\sigma}=\frac{1}{2n}\sum_{i\in
E}\operatorname{d}(i,\sigma(i))$ and
$w_{min}=\operatorname{min}\\{w_{\sigma}\mid\sigma\in H_{n}\\}$.

Let $\mathcal{F}$ be an invariant of virtual knots/links. For collections of
open curves in 3-space, the $\mathcal{F}$-spectral measure has the following
properties :

  1. 1.

It is independent of any choice of projection of the collection of open curves
in 3-space.

  2. 2.

It is not an invariant of a corresponding/approximating classical or virtual
link.

  3. 3.

It is not a topological invariant, but it is a continuous function of the
curve coordinates.

  4. 4.

As the endpoints of a collection of open curves in 3-space tend to coincide,
according to a particular $\sigma$, its $\mathcal{F}$-spectral measure tends
to the $\mathcal{F}$ invariant of the corresponding classical link in 3-space
(see Theorem 6.1).

In particular, we can define the spectral height, the spectral genus, the
spectral Jones polynomial and the spectral arrow polynomial for collections of
open curves in 3-space. The spectral height and spectral genus take values in
$\mathbb{R}$ whereas the spectral Jones polynomial and the spectral arrow
polynomial are polynomial with coefficients in $\mathbb{R}$.

######  Theorem 6.1.

Let $\mathcal{L}$ be a collection of open curves in 3-space with $n$
components and strand permutation, $\tau$ on the set of endpoints, $E$. Let
$\mathcal{F}$ be an invariant of virtual links. As the endpoints tend to
coincide according to a closure permutation, $\sigma$, the
$\mathcal{F}$-spectral measure tends to the $\mathcal{F}$-invariant of the
resultant closed link in 3-space.

###### Proof.

As endpoints tend to coincide, $w_{min}\rightarrow 0$ and
$\mathcal{F}_{\sigma}\left(({\mathcal{L}}_{\tau},\sigma)_{v}\right)$ is the
only surviving term in $\mathcal{F}(\mathcal{L}_{\tau})$. The claim follows
from Corollary 6.4. ∎

##  7 Conclusions

In this paper, new invariants of linkoids are introduced via a mapping from
linkoids to virtual links. This leads to a new collection of strong invariants
of linkoids that are independent of any given virtual closure and to novel
measures of entanglement of open curves in 3-space. These measures are
continuous functions of the curve coordinates and tend to the corresponding
classical invariants when the endpoints of the curves tend to coincide.

The mapping from linkoids to virtual links gives us access to many ideas and
invariants of virtual links that we can now apply to studying linkoids and
open curves in 3-space. Thus the program we follow here gives new applications
of virtual knot theory to classical and physical knotting.

The association of virtual knots to linkoids and open curves in three space
can be interpreted as a way to construct embedded curves in thickened surfaces
that reflect these topological structures. We have concentrated on making new
invariants of linkoids and open curves, but the geometry of the association
with curves in thickened surfaces remains to be more fully investigated in the
future.

##  8 Acknowledgements

Kasturi Barkataki and Eleni Panagiotou were supported by NSF DMS-1913180 and
NSF CAREER 2047587.

## References

  * [1] Fukuda M, Kotani M, Mahmoudi S.  Classification of doubly periodic untwisted (p,q)-weaves by their crossing number and matrices.  J Knot Theory Ramif. 2023;32(5):1.
  * [2] Liu Y, O’Keeffe M, Treacy M, Yaghi O.  The geometry of periodic knots, polycatenanes and weaving from a chemical perspective: a library for reticular chemistry.  Chem Soc Rev. 2018;47:4642-64.
  * [3] Edwards SF.  Statistical mechanics with topological constraints: I.  Proc Phys Soc. 1967;91:513-9.
  * [4] Panagiotou E, Kauffman L.  Knot polynomials of open and closed curves.  Proc R Soc A. 2020;476:20200124.
  * [5] Panagiotou E, Kauffman L.  Vassiliev measures of complexity for open and closed curves in 3-space.  Proc R Soc A. 2021;477:20210440.
  * [6] Barkataki K, Panagiotou E.  The Jones polynomial of collections of open curves in 3-space.  Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. Nov 2022;vol. 478, no. 2267.
  * [7] Jones VFR.  A polynomial invariant of knots via von Neumann algebras.  Bull Am Math Soc. 1985;12:103-12.
  * [8] Freyd P, Yetter D, Hoste J, Lickorish W, Millett KC, Ocneanu A.  A new polynomial invariant for knots and links.  Bull Am Math Soc. 1985;12:239-46.
  * [9] Jones VFR.  Hecke algebra representations of braid groups and link polynomials.  Annals of Mathematics. 1987;126:335-88.
  * [10] Kauffman LH.  State models and the Jones polynomial.  Topology. 1987;26:395-407.
  * [11] Kauffman LH.  An invariant of regular isotopy.  Trans Amer Math Soc. 1990;318:417-71.
  * [12] Przytycki J, Traczyk P.  Conway algebras and skein equivalence of links.  Proc Amer Math Soc. 1987;100:744-8.
  * [13] Turaev V.  Knotoids.  Osaka, J Math. 2012;49:195-223.
  * [14] Gügümcu N, Kauffman LH.  New invariants of knotoids.  Eur J Comb. 2017;65:186-229.
  * [15] Kauffman LH.  Virtual Knot Theory.  Europ J Combinatorics. 1999;20:663-91.
  * [16] Gauss KF.  Werke.  Kgl. Gesellsch. Wiss. Göttingen; 1877.
  * [17] Carter JS, Kamada S, Saito M.  Stable equivalence of Knots on surfaces and Virtual Knot cobordisms.  Journal of Knot Theory and Its Ramifications. 2000;11:311-22.
  * [18] Manturov VO.  Virtual crossing numbers for virtual knots.  Journal of Knot Theory and Its Ramifications. 2012;21(13):1240009.
  * [19] Kauffman LH, Lambropoulou S.  Virtual braids and the L-move.  Journal of Knot Theory and Its Ramifications. 2006;15(06):773-811.
  * [20] Satoh S, Tomiyama Y.  On the crossing numbers of a virtual knot.  Proceedings of the American Mathematical Society. 2012;140(1):367-76.
  * [21] Kuperberg G.  What is a virtual link?  Algebraic & Geometric Topology. 2003;3(1):587 591.  Available from: https://doi.org/10.2140/agt.2003.3.587.
  * [22] Dye H, Kauffman LH.  Virtual crossing number and the arrow polynomial.  Journal of Knot Theory and Its Ramifications. 2009;18(10):1335-57.
  * [23] Gügümcu N, Lambropoulou S.  Knotoids, braidoids and applications.  Symmetry. 2017;9:315.
  * [24] Gügümcü N, Kauffman LH.  Parity, virtual closure and minimality of knotoids.  Journal of Knot Theory and Its Ramifications. 2021;30(11):2150076.
  * [25] Manouras M, Lambropoulou S, Kauffman LH.  Finite type invariants for knotoids.  European Journal of Combinatorics. 2021 dec;98:103402.  Available from: https://doi.org/10.1016%2Fj.ejc.2021.103402.
  * [26] Kauffman LH.  An affine index polynomial invariant of virtual knots.  Journal of Knot Theory and Its Ramifications. 2013;22(04):1340007.
  * [27] Kauffman LH.  A self-linking invariant of virtual knots.  Fund Math. 2004;184:135-58.  Available from: https://doi.org/10.4064/fm184-0-10.
  * [28] Kaestner AM, Kauffman LH.  Parity, skein polynomials and categorification.  Journal of Knot Theory and Its Ramifications. 2012;21(13):1240011.
  * [29] Panagiotou E.  The linking number in systems with periodic boundary conditions.  J Comput Phys. 2015;300:533-73.
  * [30] Morton HR, Grishanov S.  Doubly periodic textile structures.  J Knot Theory Ramif. 2009;18:1597-622.
  * [31] Barkataki K, Panagiotou E.  The Jones polynomial in systems with Periodic Boundary Conditions.  arXiv 230914572. 2023.
  * [32] Evans ME, Robins V, Hyde ST.  Periodic entanglement II: weavings from hyperbolic line patterns.  Acta Chryst. 2013;A69:262-75.