Invariants of Multi-linkoids

In this paper, we extend the definition of a knotoid to multi-linkoids that consist of a finite number of knot and knotoid components. We study invariants of multi-linkoids, such as the Kauffman bracket polynomial, ordered bracket polynomial, the Kauffman skein module, and the T-invariant in relation with generalized Θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Theta $$\end{document}-graphs.


Introduction
Knotoids were defined by Turaev [36] as immersions of the unit interval in an arbitrary surface.They can be regarded as open-ended knot diagrams with two endpoints, on which we consider an equivalence relation induced by the Reidemeister moves.In this sense, the theory of knotoids in S 2 generalizes classical knot theory and has a natural connection with the theory of virtual knots [21,22] through the virtual closure.The intrinsic invariants of knotoids, as well as how knot invariants extend as knotoid invariants, have been studied by many researchers.See [3,4,5,6,12,13,14,16,17,23,24,34,36] for studies on knotoids.
Knotoids can be considered in a more topological setting, as planar knotoids are equivalent to relative knots in R 3 \ {two parallel lines} [12,31].In relation to this topological setting, Turaev showed that knotoids in S 2 are in one-to-one correspondence with simple Θ-curves and multi-knotoids in S 2 , immersions of the unit interval and a finite number of circles in S 2 , are in one-to-one correspondence with simple theta-links [36].
Topological structures are also an emerging field in modern chemistry [9] as knots have been identified DNA [32] and proteins [25,38].Research suggest that knots increase thermal and kinetic stability of the molecule [33], as well as have important functional roles [38].A protein's backbone natively forms an embedded interval in 3-space.Classical studies of protein topology (i.e.determining the knot type) rely on closing this interval by some closure method (closure methods are thoroughly discussed in [26]).Since closures are often ambiguous (in the case when the protein termini are not located close to the minimal convex surface enveloping the protein), knotoids have been identified as natural candidates to study open-knotted proteins [19,18,15].
One can extend the notion of a knotoid by considering several closed and open-ended components.A multi-linkoid is a union of a finite number of immersed unit intervals and circles in a closed orientable surface.
We expect that multi-linkoids would suggest a new setting for the topological analysis of several mutually entangled polymer chains or subchains via the invariants we introduce here.
The main goal of this paper is to generalize the mentioned concepts to multi-linkoids and to introduce invariants for them.
The paper is organized as follows.Section 1 is an overview of the required notions related to multi-linkoids.In Section 2 we extend the Kauffman bracket polynomial to multi-linkoids.In addition, we strengthen this invariant to the ordered Kauffman bracket polynomial for multi-linkoids with an ordering on its knotoid components.In Section 2.1 we introduce the Kauffman skein module of multi-linkoids and show that the module if freely generated.In Section 3 we study multi-linkoids in the topological setting.We show that multi-linkoids are equivalent to relative links in R 3 and introduce simple generalized Θ-graphs, which are in one-to-one correspondence with multi-linkoids, where we take also into consideration the possibility of an ordering on the components.
In Section 4 we utilize the T -invariant for spatial graphs [20] and strengthen it by introducing the colored T -invariant for distinguishing multi-linkoids and ordered multi-linkoids.

Preliminaries
Definition 1.A multi-linkoid diagram in a closed (oriented or unoriented) surface Σ is an immersion of a number of unit intervals [0, 1] and unit circles S 1 into Σ.This immersion is generic in the sense that there are only a finite number of intersections of the image that we endow each with under or over information, and regard them as crossings of the multi-linkoid diagram.
The images of the points 0 and 1 of the unit intervals are considered to be distinct from each other and are called the endpoints of the diagram.We consider an orientation on each of the components of a multi-linkoid diagram in a way that each open-ended component is oriented from the image of 0 named specifically as tail to the image of 1 named specifically as head.As in the case of knotoids [36], we consider multi-linkoids up to the equivalence relation generated by the Reidemeister moves.Definition 2. Two multi-linkoids are equivalent if and only if they differ by a finite sequence of Reidemeister moves R-0 (surface isotopy), R-I, R-II, and R-III, shown in the Figure 2. It is not allowed to move the endpoints over or under a strand as depicted in Figure 3.In [27] framed knotoids and their quantum invariants were introduced.
Definition 3. A framed knot is a knot equipped with a transversal, smooth, everywhere nonzero vector field (or equivalently, an embedding of a ribbon/annulus in the 3-space).
The move R-I changes the framing of a knotoid, we thus replace it with the move FR-I depicted in Figure 4. Framed multi-linkoids are now naturally defined by the following definition.Definition 4. A framed multi-linkoid on an orientable surface Σ is an equivalence class of knotoid diagrams under the equivalence generated by R-0, FR-I, R-II, and R-III.
In an oriented multi-linkoid diagram we assign to each crossing a sign using the convention sign = 1 and sign = −1.
Definition 5.The sum of signs over all crossings of an oriented multi-linkoid diagram of L is called the writhe, w(L), of L.
Note that moves FR-I, R-II, and R-III do not change the writhe, w(K) is thus an invariant of framed oriented multi-linkoids.The ordering on the n open components of a multi-linkoid diagram induces an ordering of the endpoints.Precisely, each endpoint is enumerated by an integer i from {1, 2, . . ., 2n}, starting from the first component with respect to the given ordering.In the sequel, some invariants of multi-linkoid diagrams are constructed with respect to the ordering induced at the endpoints, see Section 2 for the ordered Kauffman bracket, Section 2.1 for the ordered Kauffman bracket skein module and Section 4 for the invariant T col .

Kauffman bracket polynomial of multi-linkoids
In this section we extend the Kauffman bracket polynomial of knotoids to multi-linkoids in S 2 or R 2 .
Let L be a multi-linkoid diagram in S 2 with n crossings.Without any consideration of orientation on the components of L, we smooth all the crossings of L by A-type and B-type smoothings that are determined by the four local regions adjacent to the crossings.The two regions that are swept by 90 degrees rotation of the overpassing strand of a crossing in the counterclockwise direction are labeled with the letter A, and the remaining two regions are labeled with the letter B. An A-type smoothing of a crossing is to remove the crossing and connect two of the A-regions and a B-type smoothing of a crossing is to remove the crossing and connect two of the B-regions that adjacent to the crossing.See Figure 5.
The resulting collection of curves from a chosen smoothing type of each crossing of L is called a state of L and denoted by σ i where i ∈ {0, 1} n .Each state of L contains a number of simple closed curves and simple arcs in S 2 containing the endpoints of L. Each circle component is then assigned the value −A 2 − A −2 and each open component is assigned the variable λ cofactored by the product of the labels A's and B's coming from the smoothing types made to obtain σ.We take B = A −1 as in the knotoid bracket case [36], and we obtain the following Laurent polynomial in the variables A, A −1 for L.
where < L | σ > is the product of the labels of the state σ i , ||c|| is the number of closed components of σ i and ||l|| is the number of open components of σ i .
In the case of a multi-knotoid diagram, we obtain the usual Kauffman bracket polynomial by taking λ = 1.
Proposition 1.The bracket polynomial is an invariant of framed multi-linkoids.The normalization is an invariant of multi-linkoids.
Proof.Since the Reidemeister moves take place locally away from endpoints, the proof of the invariance runs similarly with the case of invariance of the bracket polynomial of knotoids under R-II and R-III moves, as given in [36].Here we illustrate how the bracket polynomial changes under a R-I move for a knotoid diagram D. It is clear that the writhe of the given diagram, when considered to be oriented, is increased by one under the R-I move.Therefore, the multiplicative factor −A 3 added by the R-I move in the bracket polynomial can cancelled by the factor (−A 3 ) −w(D)+1 , where −w(D) + 1 is clearly the writhe of the resulting diagram by the R-I move.See Figure 6.Let us consider an ordered version of the Kauffman bracket polynomial.Let L be an ordered multi-linkoid diagram in S 2 .We enumerate each of its endpoints by an integer i ∈ {1, . . ., 2n}, n ≥ 1, according to the ordering on the open components of L. We apply the bracket smoothing at each crossing of L to obtain the states of L. Each open component of a state is assigned λ ij , i < j where i, j are the integer labels at the endpoints of the components.Circle components of a state are again assigned the value −A 2 − A −2 .We obtain the following Laurent polynomial in variables A, A −1 , λ ij (i < j).Definition 8.The ordered bracket polynomial of L, < L > • is defined to be the sum where the sum is taken over all states of L, < K | σ > is the product of the smoothing labels of a state σ and Λ is the collection of open components in the state σ.
Note that if L is a multi-knotoid diagram, the ordering on L is trivial since there is only one knotoid component of L.Then, the ordered bracket polynomial of L can be assumed to be equal to the Kauffman bracket polynomial of L with λ 12 = λ = 1.
Proposition 2. The ordered bracket polynomial is an framed ordered multi-linkoid invariant.The normalization of the ordered bracket polynomial, is an invariant of ordered multi-linkoids.
Proof.It is clear that the Reidemeister moves FR-I, R-II, and R-III preserve both the framing and ordering on a multi-linkoid diagram, and the invariance of the polynomial under these moves follows similarly as the invariance of the bracket polynomial.The ordered bracket polynomial behaves the same under an R-I move as the bracket polynomial, which implies that the normalization by the factor (−A 3 ) −w(L) becomes an ordered multi-linkoid invariant.

The Kauffman bracket skein module
Observe that in the bracket polynomial formulas ( 1) and ( 2), we only consider the number of closed components ||c|| in each state, where each component is assigned the term −A 2 − A −2 .We can extend the bracket polynomial so that in each state we also keep information about the homology classes of elements in c in the complement S 2 \ l.We do this by introducing the Kauffman bracket skein module (KBSM) of multi-linkoids.Furthermore, we extend this invariant to multi-linkoids in any closed, connected, orientable genus g surface.Skein modules were independently introduced by Turaev [37] and Przytycki [31], they can be viewed as generalizations of invariants based on the skein relation for knots in 3-manifolds.The idea behind our construction (which is closely related to the original construction) is that we first construct a space of all possible linear combinations of multi-linkoids and in this space impose the skein and framing relation, which characterize the bracket polynomials given by formulas (1) and (2).For similar constructions see [8,7,11,10,30].Definition 9. Let Σ be a closed connected orientable surface of genus g.Let R be a commutative ring with an invertible element A (e.g. the ring of Laurent polynomials Z[A, A −1 ]) and let L fr Σ,n be the set of framed multi-linkoids on Σ with 2n endpoints.Denote by R[L fr Σ,n ] the free R-module spanned by L fr Σ,n and by S(L fr Σ,n , R, A) the submodule of R[L fr Σ,n ] generated by the following two expressions (relators): ) represent classes of multi-linkoids that are everywhere the same except inside a small disk where they look like the figures indicated.
The Kauffman bracket skein module of multi-linkoids in Σ with 2n endpoints is the quotient module e. all formal finite linear sums of multi-linkoids in which we enforce the two relations obtained by the expressions (3).Theorem 1.Let B fr Σ,n be the set of all multi-linkoids in Σ with 2n endpoints without crossings and without trivial contractible components.The module K fr Σ,n (R, A) is freely generated by B fr Σ,n , i.e.
Proof.In order to prove the theorem, it is enough to show that B fr Σ,n is a basis for K fr Σ,n (R, A) and that for a given multi-linkoid L ∈ L fr Σ,n the expression [L] ∈ K fr Σ,n (R, A), written in terms of the basis, is unique.
For a given representative L of [L] ∈ K fr Σ,n (R, A), we can first remove all crossings using (3a) and then remove all trivial components using (3b), we end up with a formal linear sum of elements B fr Σ,n .Elements B fr Σ,n thus generate of the module.To show that the expression [L], evaluated in B fr Σ,n , is unique, we need to show that it does not depend on any choice we can make during the computation of [L] and that it is invariant under Reidemeister moves.We enumerate the crossings of L by ordinals 1, 2, . . ., k.We first show that the result does not depend on the order we perform crossing eliminations via (3a).

Let i j
represent a multi-linkoid with crossings i and j marked.The result does not depend on the order of crossings smoothings: In addition, the result does not depend on the elimination order of trivial components using the framing relation (3b).
Proving invariance is similar to that in the classical case.The expression is invariant under Reidemeister move R-II: where we used the framing relation (3b) in the last equality.The expression is also invariant under Reidemeister move R-III: where the second equality holds by invariance under R-II.
Let L be a multi-linkoid in Σ with 2n endpoints.We denote by written in terms of elements in the basis B fr Σ,n (or just [L] if we fix the basis and the ambient space is known from context).Due to Theorem 1, [L] is an invariant of ordered framed multi-linkoids.
As in the classical case, we can obtain an invariant of non-framed linkoids by multiplying it by (−A 3 ) −w(L) .The expression is an invariant of multi-linkoids in Σ.
Let us now consider the ordered case.For our construction, we will need the set of multi-linkoids with arbitrary ordering on the vertices (not necessarily consecutive on the endpoints on the same component), i.e. each endpoint is assigned exactly one value in {1, 2, . . ., 2n}.We call such multilinkoids vertex-ordered multi-linkoids.
Let the set Lfr Σ,n of all vertex-ordered multi-linkoids with n open components and repeat the construction.The quotient module is the Kauffman bracket skein module of vertex-ordered multi-linkoids in Σ with 2n endpoints.As ordered multi-linkoids are just special cases of vertex-ordered multi-linkoids, we will obtain an ordered multi-linkoids invariant through Kfr Σ,n (R, A).
Theorem 2. Let Bfr Σ,n the set of all isotopy classes of vertex-ordered multi-linkoids in Σ with 2n endpoints without crossings and without trivial contractible components.The module Kfr Σ,n (R, A) is freely generated by the basis Bfr Σ,n , i.e.
Proof.The proof is essentially the same to that of Theorem 1.We can reduce every link L ∈ Lfr Σ,n to a formal sum of elements from Bfr Σ,n by operations (3).Clearly, the result also does not depend on the order we perform the operations.To show invariance under Reidemeister moves, we repeat the argument from the proof of Theorem 1 verbatim.
As before, the normalized expression is an invariant of vertex-ordered multi-linkoids on Σ and, as a special case, an invariant of ordered multi-linkoids on Σ.
The following propositions follow directly from construction.Proposition 4. Given an ordered multi-linkoid L in S 3 (or R 3 ), the ordered Kauffman bracket polynomial by replacing each open component with endpoints, enumerated by i and j, by λ i,j and replacing each closed component with the factor Example 2. Consider the two oriented multi-linkoids K 1 2 in R in Figure 8.We have A straightforward computation shows us:  The normalized ordered Kauffman bracket polynomial does not distinguish the two multi-linkoids: Observe that if we consider that the two multi-knotoids lie in the S 2 , K 1 and K 2 are isotopic.Indeed, 3 Spatial graphs and multi-linkoids Similar to knotoids in R 2 , we can consider the topological setting of multi-linkoids in R 2 .A planar multi-linkoid can be uniquely lifted in the 3-space as a relative link in R 3 \ {2n parallel lines}.See Figure 9.By contracting these lines at ±∞, they can be presented as spacial cases of generalized Θ-graphs, which we will be the topic of this section.A spatial graph diagram is a regular projection of a graph on S 2 ⊂ S 3 endowed under or over information.
Theorem 3 (Kauffman [20]).Two diagrams of spatial graphs are ambient isotopic if and only if they are related through a finite sequence of moves R-0, R-I, R-II, R-III involving the edges (Figure 2) and moves R-IV and R-V involving the vertices (Figure 10).Definition 11.A Θ-graph is a spatial graph with two labeled vertices v 0 , v 1 and three labelled edges e + , e 0 , e − connecting the vertices.Two Θ-graphs are considered to be isomorphic if there is an orientation preserving isotopy of S 3 between them that preserves the labelling of the vertices and the edges.
One can work with Θ-graphs through their diagrams in S 2 ⊂ S 3 .We assume two Θ-graph diagrams represent the same Θ-graph in S 3 if they are related to each other by a sequence of spatial graph Reidemeister moves, presented in Figure 2b.A knotoid diagram in S 2 can be assigned to a simple Θ-graph embedded in S 3 by corresponding its endpoints to two vertices, and the knotoid diagram is placed as the edge labeled by e 0 of the Θ-graph.This construction was presented in [36], where it was proven that it induces a bijection between the set of isotopy classes of spherical knotoids and the isomorphism classes of simple Θgraphs.In [36], this bijection was used to prove the prime decomposition theorem of knotoids.Now we shall generalize the concept of a Θ-graph and give a correspondence between linkoids in S 2 and the generalized Θ-graphs.In this way, we extract invariants for linkoids from spatial graph invariants.
Definition 13.A generalized Θ-graph is a connected graph embedded in S 3 with an even number, say 2n, n ∈ N, of trivalent vertices, labeled v i and w i where i ∈ {1, . . ., n}, and exactly two vertices v ∞ , v −∞ that are located at N (0, 0, 0, 1) ∈ S 3 and S(0, 0, 0, −1) ∈ S 3 , respectively.The edge set E(G) consists of edges {v i w i } n i=1 , edges {v i v ∞ } n i=1 and edges {v i v −∞ } n i=1 connecting v i and the vertices v ∞ and v −∞ , and edges {w i v ∞ } n i=1 and {w i v −∞ } n i=1 connecting w i and the points v ∞ and v −∞ .Note that each vertex of a pair (v i , w i ) is adjacent to both v ∞ and v −∞ .Definition 14.A generalized Θ-graph is simple if for every i, j ∈ {1, . . ., n}, the subgraph induced by edges Let L be a multi-linkoid diagram in S 2 with 2n endpoints.We assign to L to a generalized Θ-graph in embedded in S 3 , which we denote by Θ(L).The graph Θ(L) is constructed as follows.We label each component of L by i ∈ {1, . . ., n}, and denote by v i the tail and by w i and the head of component i, such that component i corresponds to an edge v i w i in Θ(L).We also add the two vertices v ∞ and v −∞ located at N (0, 0, 0, 1) ∈ S 3 and S(0, 0, 0, −1) ∈ S 3 , respectively.
Next, we add to the edge set of Θ(L) the edges v i v ∞ , v i v −∞ , w j v ∞ , and w j v −∞ , such that they form a graph C 4 that bounds a disk and the arcs v i v ∞ and w i v ∞ form only over-crossings with the rest of the diagram and arcs v i v −∞ and w i v −∞ form only under-crossings with the rest of the diagram.
The assigned graph Θ(L) is a simple generalized Θ-graph that contains 2n + 2 vertices, 2n of them are of degree 3, and the two vertices at the poles are of degree 2n.
Figure 11: A linkoid and the corresponding generalized Θ-graph.
Theorem 4.There is a one-to-one correspondence between the set of all multi-linkoids in S 2 and the set of simple generalized Θ-graphs.
Proof.It is clear that any of the R-I, R-II, R-III moves takes place locally away from the endpoints of a multi-linkoid diagram, and they transform to either one of the R-I, R-II, R-III moves of the spatial graph, or a combination of them, in case when the arcs, connecting vertices labeled by v i , w j to the infinity vertices, conflict with the local move region on the corresponding generalized θ-graph diagram.The isotopy moves that displace endpoints of a multi-linkoid diagram transform as a combination of spatial Reidemeister moves including R-IV and R-V-moves on the corresponding generalized θ-graph diagram.See Figure 12 for some of the instances.

A colored version of Kauffman's T invariant
In [20] an invariant T of spatial graphs in S 3 was introduced.For convenience, we repeat the construction.
The special cases of multilinkoids are knotoids consisting of only one open (knotoid) component and multi-knotoids consisting of only one open component and a number of closed (knot) components.

Figure 1 :
Figure 1: A multi-linkoid diagram with three components on the torus.

Figure 5 :
Figure 5: Smoothing types of a crossing

Figure 6 :
Figure 6: Behaviour of the Kauffman bracket polynomial under a R-I move.

Figure 7 :
Figure 7: Two ordered linkoids that are distinguished by the ordered bracket polynomial

Proposition 3 .
Given an (unordered) multi-linkoid L in S 3 (or R 3 ), the Kauffman bracket polynomial < L > (A, λ) is obtained from [L] by replacing each open component with λ and every closed component by the factor (−A 2 − A −2 ).

Figure 10 :
Figure 10: Local Reidemeister moves for spatial graphs involving a vertex.The move R-IV can have an arbitrary number of strands (including zero) on the left-hand and the right-hand side of the diagram.The move R-V can have an arbitrary number of strands on the left-hand side of the diagram.

Definition 12 .
A Θ-graph is called simple if the edges e + and e − bound a 2-disk.