aa r X i v : . [ m a t h . G T ] J a n PSEUDO LINKS AND SINGULAR LINKS IN THE SOLID TORUS
IOANNIS DIAMANTIS
Abstract.
In this paper we introduce and study the theories of pseudo links and singularlinks in the Solid Torus, ST. Pseudo links are links with some missing crossing informationthat naturally generalize the notion of knot diagrams, and that have potential use in molecularbiology, while singular links are links that contain a finite number of self-intersections. Weconsider pseudo links and singular links in ST and we set up the appropriate topological theoryin order to construct invariants for these types of links in ST. In particular, we formulate andprove the analogue of the Alexander theorem for pseudo links and for singular links in ST. Wethen introduce the mixed pseudo braid monoid and the mixed singular braid monoid, with theuse of which, we formulate and prove the analogue of the Markov theorem for pseudo links andfor singular links in ST.Moreover, we introduce the pseudo Hecke algebra of type A, P H n , the cyclotomic andgeneralized pseudo Hecke algebras of type B, P H ,n , and discuss how the pseudo braid monoid(cor. the mixed pseudo braid monoid) can be represented by P H n (cor. by P H ,n ). This is thefirst step toward the construction of HOMFLYPT-type invariants for pseudo links in S and inST. We also introduce the cyclotomic and generalized singular Hecke algebras of type B, S H ,n ,and we present two sets that we conjecture that they form linear bases for S H ,n . Finally, wegeneralize the bracket polynomial for pseudo links in ST.2010 Mathematics Subject Classification. Introduction
Pseudo diagrams of knots were introduce by Hanaki in [22] as projections on the 2-spherewith over/under information at some of the double points. They comprise a relatively new andimportant model for DNA knots, since there exist DNA knots that is impossible to understanda positive from a negative crossing, even when studying them by electron microscopes. Byconsidering equivalence classes of pseudo diagrams under equivalence relations generated by aspecific set of Reidemeister moves, one obtains the theory of pseudo knots , that is, standardknots whose projections contain crossings with missing information (see [23] for more details).In [6], the pseudo braid monoid is introduced, which is related to the singular braid monoid,and with the use of which, the authors present the analogues of the Alexander and the Markovtheorems for pseudo knots in S . From now on and throughout the paper, by pseudo links weshall mean both pseudo links and pseudo knots. Moreover, it is worth mentioning that in [8], theauthor introduces the L -moves for pseudo links and formulates and proves a sharpened versionof the analogue of the Markov theorem for pseudo links, as well as an alternative proof of theanalogue of the Alexander theorem for pseudo links in S . In this paper we extend these resultsfor pseudo links in the solid torus ST. Our aim is the construction of pseudo knot invariantsin S and ST, and toward that end we present the pseudo bracket polynomial for pseudo links Key words and phrases. solid torus, pseudo knots, pseudo links, singular knots, mixed pseudo links, mixedsingular links, mixed pseudo braids, mixed pseudo braid monoid, mixed singular braids, pseudo braid monoid oftype B, mixed singular braid monoid, pseudo Hecke algebra of type A, pseudo Hecke algebras of type A and oftype B, cyclotomic and generalized singular Hecke algebras of type B, pseudo bracket polynomial. n ST and discuss HOMFLYPT-type invariants for pseudo links in S and ST via appropriateHecke type algebras, following [25, 28, 33].Singular knots are knots with finite many self-intesections, and as shown in [6], the theoryof singular links in S is close related to the theory of pseudo links in S . In this paper, weestablish this correspondence for the case of ST, and through this correspondence we formulatethe analogues of the Alexander and the Markov theorems for singular links in ST. Then, following[33], we define the generalized (and cyclotomic) singular Hecke algebras of type B and discussfurther research needed toward the construction of HOMFLYPT-type invariants for singularlinks in ST, which is the subject of a sequel paper.The paper is organized as follows: In § § § S , and we present some results toward theconstruction of HOMFLYPT-type invariants for pseudo links in S . In particular, we presenta spanning set for the pseudo Hecke algebra of type A. Moreover, we introduce the cyclotomicand the generalized pseudo Hecke algebras of type B, and which are related to the pseudo knottheory of ST and, as in the type A case, we discuss further steps toward the construction ofHOMFLYPT-type invariants for pseudo links in ST. We conclude by presenting potential basesfor these algebras. In a similar way, we also introduce the cyclotomic and generalized singularHecke algebras of type B, through which, HOMFLYPT-type invariants for singular links in STmay be constructed. Finally, in § S . Our aim is to extend theseinvariants for the case of lens spaces L ( p, q ) following [16, 17, 18].1. Preliminaries
The theories of pseudo links and singular links in S . In this subsection we recallresults on pseudo links and on singular links in S . In particular, we recall the pseudo braidmonoid, P M n , introduced in [6], and that is related to the singular braid monoid, SM n [3, 4],and recall all necessary results in order to present the analogues of the Alexander and the Markovtheorems for pseudo links and singular links in S .A pseudo diagram of a knot consists of a regular knot diagram with some missing crossinginformation, that is, there is no information about which strand passes over and which strandpasses under the other. These undetermined crossings are called pre-crossings (for an illustrationsee Figure 1). Definition 1.
Pseudo knots are defined as equivalence classes of pseudo diagrams under anappropriate choice of Reidemeister moves that are illustrated in Figure 2.As explained in [6], pseudo knots are closed related to singular knots , that is, knots thatcontain a finite number of self-intersections. In particular, there exists a bijection f from thesingular knot diagrams to the set of pseudo knot diagrams where singular crossings are mappedto pre-crossings. In that way we may also recover all of the pseudo knot Reidemeister moves, igure 1. A pseudo knot.
Figure 2.
Reidemeister moves for pseudo knots.with the exception of the pseudo-Reidemeister I (PR1) move (see Figure 2). Moreover, f inducesan onto map from singular knots to pseudo knots, since the image of two isotopic singular knotdiagrams are also isotopic pseudo knot diagrams with exactly the same sequence of Reidemeistermoves.We now introduce the pseudo braid monoid, P M n , following [6]. Definition 2.
The monoid of pseudo braids,
P M n , is the monoid generated by σ ± i , p i , i =1 , . . . , n −
1, illustrated in Figure 3, where σ ± i generate the braid group B n and p i satisfy thefollowing relations: i. p i p j = p j p i , if | i − j | ≥ ii. p i σ ± j = σ ± j p i , if | i − j | ≥ iii. p i σ ± i = σ ± i p i , i = 1 , . . . , n − iv. σ i σ i +1 p i = p i +1 σ i σ i +1 , i = 1 , . . . , n − v. σ i +1 σ i p i +1 = p i σ i +1 σ i , i = 1 , . . . , n − p i corresponds to a standard crossing and not to a singular crossing. We denotea singular crossing by τ i and we have that if we replace the pre-crossings p i of Definition 2 by igure 3. The pseudo braid monoid generators.singular crossings τ i , we obtain the singular braid monoid, SM n , defined in [3, 4]. Thus, weobtain the following result: Proposition 1 (Proposition 2.3 [6]) . The monoid of pseudo braids is isomorphic to the singularbraid monoid, SM n . Remark 1.
In [20] it is shown that SM n , the singular braid monoid, embeds in a group, thesingular braid group SB n . It follows that P M n embeds in a group also, the pseudo braid group P B n , generated by σ i and p i , i = 1 , . . . , n −
1, satisfying the same relations as
P M n . Obviously, SB n is isomorphic to P B n (recall Proposition 2.3 [6]).Define now the closure of a pseudo braid (cor. of a singular braid) as in the standard case (foran illustration of the closure of a pseudo braid see Figure 8). By considering P M n ⊂ P M n +1 ,we can consider the inductive limit P M ∞ . Using the analogue of the Alexander theorem forsingular knots ([4]), in [6] the analogue of the Alexander theorem for pseudo links is presented.In particular: Theorem 1 ( Alexander’s theorem for pseudo links).
Every pseudo link can be obtainedby closing a pseudo braid.
We now recall the notion of L -moves for pseudo knots and results from [8, 27]. L -movesmake up an important tool for braid equivalence in any topological setting and they allow usto formulate sharpened versions of the analogues of the Markov theorems. For more details thereader is referred to [27] and references therein. Definition 3. An L -move on a braid β , consists in cutting an arc of β open and pulling theupper cut-point downward and the lower upward, so as to create a new pair of braid strandswith corresponding endpoints (on the vertical line of the cut-point), and such that both strandscross entirely over or under with the rest of the braid. Stretching the new strands over will giverise to an L o -move and under to an L u -move as shown in Figure 4 by ignoring all pre-crossings. L -moves for pseudo braids are defined in the same way as in the singular case, that is, the twostrands that appear after the performance of an L -move should cross the rest of the braid onlywith real crossings (all over in the case of an L o -move or all under in the case of an L u -move).For an illustration see Figure 4 for the case of an L -move performed on a pseudo braid.As explained in [8], L -moves on pseudo knots can be used in order to obtain the analogue ofthe Alexander theorem for pseudo knots following the braiding algorithm in [4], [1] and [30]. Themain idea of this algorithm is to keep the arcs of the oriented links diagram that go downwardswith respect to the height function unaffected and replace arcs that go upwards with braidstrands. We first isotope the pre-crossings in such a way that the braiding algorithm will notaffect them (see Figure 10), and we may then apply the braiding algorithm of [4] or [30] by igure 4. L -moves for pseudo braids.ignoring the pre-crossings. The same is true for the case of singular links in S (for a detailedproof for the case of singular links in S see [33] Theorem 2.3). In § S ([21, 27, 8, 6]). Theorem 2 ( The analogue of the Markov Theorem for singular links).
Two orientedsingular links are isotopic if and only if any two corresponding singular braids differ by braidrelations in SB ∞ and a finite sequence of the following moves: Singular commuting : τ i α ∼ α τ i L − moves, where α ∈ SB n , the singular braid group, and τ i a singular crossing. Theorem 3 ( The analogue of the Markov Theorem for pseudo braids).
Two pseudobraids have isotopic closures if and only if one can be obtained from the other by a finite sequenceof the following moves:
Conjugation : α ∼ β ± α β ∓ , for α ∈ P M n & β ∈ B n ,Commuting : α β ∼ β α, for α, β ∈ P M n ,Stabilization : α ∼ α σ ± n , for α ∈ P M n ,P seudo − stabilization : α ∼ α p n , α ∈ P M n . Equivalently, two pseudo braids have isotopic closures if and only if one can be obtained fromthe other by a finite sequence of the following moves: L − movesCommuting : α β ∼ β α, for α, β ∈ P M n ,P seudo − Stabilization : α ∼ α p n , ∈ P M n +1 . The knot theory of ST.
We now view ST as the complement of a solid torus in S and we present results from [30]. An oriented link L in ST can be represented by an oriented mixed link in S , that is, a link in S consisting of the unknotted fixed part b I representing thecomplementary solid torus in S and the moving part L that links with b I . mixed link diagram is a diagram b I ∪ e L of b I ∪ L on the plane of b I , where this plane isequipped with the top-to-bottom direction of I . For an illustration see Figure 6 by ignoring thepre-crossings.Consider now an isotopy of an oriented link L in ST. As the link moves in ST, its correspondingmixed link will change in S by a sequence of moves that keep the oriented b I pointwise fixed.This sequence of moves consists in isotopy in the S and the mixed Reidemeister moves . Interms of diagrams we have the following result for isotopy in ST:The mixed link equivalence in S includes the classical Reidemeister moves and the mixedReidemeister moves, which involve the fixed and the standard part of the mixed link, keeping b I pointwise fixed.By the Alexander theorem for knots in solid torus ([29]), a mixed link diagram b I ∪ e L of b I ∪ L may be turned into a mixed braid I ∪ β with isotopic closure. This is a braid in S where,without loss of generality, its first strand represents b I , the fixed part, and the other strands, β ,represent the moving part L . The subbraid β shall be called the moving part of I ∪ β . For anillustration see Figure 8 by ignoring the pre-crossings.The sets of braids related to ST form groups, the Artin braid groups type B, denoted B ,n ,with presentation: B ,n = * t, σ , . . . , σ n − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) σ tσ t = tσ tσ tσ i = σ i t, i > σ i σ i +1 σ i = σ i +1 σ i σ i +1 , ≤ i ≤ n − σ i σ j = σ j σ i , | i − j | > + , where the generators σ i and t are illustrated in Figure 5. Figure 5.
The generators of B ,n .We finally have that isotopy in ST is translated on the level of mixed braids by means of thefollowing theorem. Theorem 4 ( The analogue of the Markov Theorem for mixed braids
Theorem 3, [28]) . Let L , L be two oriented links in ST and let I ∪ β , I ∪ β be two corresponding mixed braidsin S . Then L is isotopic to L in ST if and only if I ∪ β is equivalent to I ∪ β in ∞ ∪ n =1 B ,n by the following moves: ( i ) Conjugation : α ∼ β − αβ, if α, β ∈ B ,n . ( ii ) Stabilization moves : α ∼ ασ ± n ∈ B ,n +1 , if α ∈ B ,n . . Pseudo Links & Singular Links in ST
In this section we introduce and study pseudo links and singular links in ST, through mixedpseudo (cor. singular) braids and the mixed pseudo (cor. singular) braid monoid of type B.We conclude this section by formulating and proving the analogues of the Alexander and theMarkov theorems for pseudo links and singular links in ST.2.1.
Mixed pseudo links and mixed singular links.
As explained in § S and thus, pseudo links in ST can be seen as mixed pseudolinks in S containing the complementary ST. Similarly, singular links in ST can be viewed as mixed singular links in S . For an illustration of a mixed pseudo link see Figure 6, where bychanging pre-crossings to singular crossings, one obtains a singular mixed link. Figure 6.
A mixed pseudo link in S .Isotopy for pseudo links in ST is now translated on the level of mixed pseudo links in S bymeans of the following theorem: Theorem 5 ( Reidemeister’s theorem for mixed pseudo links).
Two mixed pseudo linksin S are isotopic if and only if they differ by a finite sequence of the classical and the pseudoReidemeister moves illustrated in Figure 2 for the standard part of the mixed pseudo links,and moves that involve the fixed and the standard part of the mixed pseudo links, called mixedReidemeister moves, and are illustrated in Figure 7. Figure 7.
Mixed Reidemeister moves.
Remark 2.
Similarly, if we exclude PR1-moves of Figure 2 for the standard part of the mixedsingular links, and if we change the pre-crossing of Figure 7 to a singular crossing, we obtainthe analogue of the Reidemeister theorem for singular links in ST. t is worth mentioning that we do not allow special crossings, i.e. pre-crossings for the caseof pseudo links and singular crossings for the case of singular links, between the fixed and themoving part of the mixed pseudo (cor. singular) braid.2.2. Mixed pseudo braids, mixed singular braids and a braiding algorithm.
We nowdefine the mixed pseudo braids, which are related to mixed pseudo links, and the mixed singularbraids, related to the mixed singular links.
Definition 4. A mixed pseudo braid (cor. mixed singular braid) on n strands, denoted by I ∪ B , is an element of the pseudo braid monoid P M n (cor. singular braid monoid SM n )consisting of two disjoint sets of strands, one of which is the identity braid I representing thecomplementary solid torus in S , and n strands form the moving subbraid β representing thepseudo (cor. singular) link L in ST. For an illustration see the left hand side of of Figure 8.Moreover, a diagram of a mixed pseudo braid is a braid diagram projected on the plane of I .Without loss of generality, we assume that the first strand of the mixed pseudo braid representsthe complementary solid torus in S . This can be realized by performing the technique of(standard) parting, in order to separate the endpoints of the mixed braids into two differentsets, the first would represent the complementary solid torus in S (i.e., the fixed part I ), andthe last n would represent the moving part of the mixed braid, representing the link in ST, andso that the resulting braids have isotopic closures. This can be realized by pulling each pair ofcorresponding moving strands to the right and over or under the strand of I that lies on theirright according to its label. We start from the rightmost pair respecting the position of theendpoints. For more details the reader is referred to [30, 14].We also define the closure of a mixed pseudo (cor. singular) braid. Definition 5.
The closure C ( I ∪ B ) of a mixed pseudo braid (cor. mixed singular braid) I ∪ B in ST is defined as in the case of classical braids in S . See Figure 8 for the case of a mixedpseudo braid. Figure 8.
The closure of a mixed pseudo braid to a mixed pseudo link.We now present a braiding algorithm for mixed pseudo links in ST and for mixed singularlinks in ST. By “mixed links” in the braiding algorithm that follows, we shall mean both mixedpseudo links and mixed singular links, and by special crossings we shall mean pre-crossings orsingular crossings. The main idea of the braiding algorithm is to keep the arcs of the orientedmixed link diagrams that go downwards with respect to the height function unaffected, andreplace arcs that go upwards with braid strands. These arcs are called up-arcs (see Figure 9). igure 9. Up-arcs.Note that the braiding algorithm should keep the fixed part of the mixed link unaffected andshould also ”take care“ of the special crossings in the mixed diagram which contain at least oneup-arc. For this we apply the idea used in [4] for the case of singular knots, (see also [26] forthe case of virtual knots). Namely, before we apply the braiding algorithm we have to isotopethe mixed link in such a way that the special crossings will only contain down-arcs, so that thebraiding algorithm will not affect them. This is achieved by rotating all special crossings thatcontain at least one up-arc, so that the two arcs are now directed downward. This is illustratedin Figure 10 for the case of pre-crossings. Then we may apply the braiding algorithm of [29],which is a modification of the braiding algorithm for oriented mixed links presented in [30], forthe mixed link (ignoring the special crossings). In particular: • We first ”take care“ of the special crossings by isotoping the mixed link in such a waythat the braiding algorithm will not affect them. For an illustration see Figure 10.
Figure 10.
Rotating pre-crossings.We now recall the braiding algorithm of [29] (for more details the reader is referredto [29] and references therein). • Consider now a vertical line l that passes through the maximum and minimum of ˆ I asillustrated in Figure 11. By small perturbations we can assume that l does not passthrough any crossings of the link. • We now apply the following braiding algorithm to the part of the link that lies to theleft of the vertical line l , keeping ˆ I fixed: – We chose a base-point and we run along the diagram of the link according to itsorientation. – When/If we run along an opposite arc, we subdivide it into smaller arcs, eachcontaining crossings of one type only as shown in Figure 9. – We now label every up-arc with an “o”or a “u”, according to the crossings it con-tains. If it contains no crossings, then the choice is arbitrary. – We perform an o -braiding moves on all up-arcs which were labeled with an “o” and u -braiding moves on all up-arcs which were labeled with an “u” (see Figure 12). igure 11Figure 12. Braiding moves for up-arcs. • We close the braided part of the link that lies to the left of l and consider this operationto be enclosed in a tube T (see left hand side of Figure 13). • We apply once again the braiding algorithm on the right hand side of l keeping ˆ I fixed,and we close now this braided portion of the link and enclose the strings participatingin this operation in a tube T (see right hand side of Figure 13). • We now rotate around the back of the diagram in order to bring the tube T to the veryright of the diagram and T to the very left of the diagram, and so that the resultingdiagram goes around a central point, say P on l . • Local maxima and minima in the diagram, if any, would lie on the vertical line l . Weisolate each one in neighborhoods that do not contain other parts of the diagram, andwe stretch the arcs above and below a point of symmetry P , so that the extrema wouldlie on l again but in reverse order. Note also that we numerate the extrema with integerswith respect to the point P . For an illustration see right hand side of Figure 13. • We open the braided diagram by cutting through a half-line starting from P and weisotope in ST. • The result is a mixed pseudo braid (cor. a mixed singular braid) whose closure is isotopicto the initial mixed link.This braiding algorithm provides a proof of the following theorem: igure 13 Theorem 6 ( The analogue of the Alexander theorem for pseudo links & for singularlinks in ST).
Every oriented mixed pseudo link is isotopic to the closure of a mixed pseudobraid and very oriented mixed singular link is isotopic to the closure of a mixed singular braid.
Remark 3. i. The braiding algorithm described above can be also applied for the case of tied pseudo links in ST, that is, pseudo links in ST equipped with some non-embeddedarcs called ties. The reader is referred to [8] for a treatment of tied pseudo links in S and to [9] for the case of tied links in various 3-manifolds. Note that tied links in S were introduced in [2] and that in [19] the author generalized the notion of tied links inST.ii. It is also worth mentioning that the [29]-braiding algorithm with the modifications de-scribed above, can be applied for braiding pseudo singular links in S and ST. Pseudosingular links are defined as links with finitely many self-intersections and with somemissing crossing information. This theory will be studied in a sequel paper for both S and ST.2.3. The mixed pseudo braid monoid & the mixed singular braid monoid of type B.
We now study algebraic structures related to mixed pseudo braids and mixed singular braids.Our aim is to formulate the analogue of the Markov theorem algebraically. We start by definingthe mixed pseudo braid monoid
P M ,n , the counterpart of the Artin’s braid group of type B forpseudo links, and also derive the mixed singular braid monoid .In order to define the mixed pseudo braid monoid P M ,n , we need to find the defining relationsof this monoid. We analyze the moves of Theorem 5 and this leads to the following definition: Definition 6.
The mixed pseudo braid monoid of type B
P M ,n is defined as the monoid gener-ated by the standard braid generators σ ± i ’s of B n , the pseudo generators p i ’s of P M n and thelooping generator t ’s of B ,n , satisfying the following relations: i σ j = σ j σ i , for | i − j | > ,σ i σ j σ i = σ j σ i σ j , for | i − j | = 1 ,p i p j = p j p i , for | i − j | > ,σ i p j = p j σ i , for | i − j | > ,σ i σ j p i = p j σ i σ j , for | i − j | = 1 ,t σ i = σ i t, for i > ,t p i = p i t, for i > ,t σ t σ = σ t σ t,t σ t p = p t σ t,σ σ − = t t − = 1 . Note that every isotopy can be decomposed in a sequence of elementary isotopies whichcorrespond to the relations in Definition 6, and that means that we have obtained a completeset of relations. It is also worth mentioning that by replacing the pre-crossings p i in Definition 6,we obtain the singular braid monoid of type B , SM ,n , presented in Theorem 4.4 [34]. Note alsothat in [34], the singular braid monoid of type B is called singular braid monoid of the Annulus.Thus, we have the following result: Theorem 7.
There exists an isomorphism µ from the singular braid monoid of type B to thepseudo braid monoid of type B, defined as follows:(1) µ : SM ,n → P M ,n σ ± i σ ± i t ± t ± τ i p i Moreover, by considering the natural inclusion
P M ,n ⊂ P M ,n +1 (see Figure 14), we mayconsider the inductive limit P M , ∞ , and similarly, by the natural inclusion SM ,n ⊂ SM ,n +1 ,the inductive limit SM , ∞ is well-defined.2.4. The analogue of the Markov theorem for mixed pseudo braids & mixed singularbraids.
Using the braiding algorithm presented in § S , we have a strongerversion of the analogue of the Markov theorem for pseudo knots, similar to Theorem 2 in [29].In particular, we have the following: Theorem 8 ( Relative version of the analogue of the Markov theorem for pseudolinks).
Two pseudo links containing the same braided part are isotopic if and only if any twocorresponding pseudo braids, both containing the same braided part, differ by conjugation & igure 14. The natural inclusion
P M ,n ⊂ P M ,n +1 . commuting, stabilization & pseudo-stabilization moves that do not affect the already braidedpart. Since all pseudo links related to the solid torus contain the same braided part ˆ I , we havethat corresponding pseudo braids contain the same braided part I . Therefore, we obtain thefollowing: Corollary 1.
Two pseudo links in ST are isotopic if and only if any two corresponding pseudobraids of theirs, differ by conjugation & commuting, stabilization & pseudo-stabilization movesthat do not affect I . By Corollary 1 and the discussion in § Theorem 9 ( The analogue of the Markov Theorem for mixed pseudo braids).
Twomixed pseudo braids have equivalent closures if and only if one can obtained from the other by afinite sequence of the following moves:
Commuting : α p i ∼ p i α, for all α ∈ P M ,n , Conjugation : β ∼ α ± β α ∓ for all β ∈ P M ,n & α ∈ B ,n , Real − Stabilization : α ∼ α σ ± n , for all α ∈ P M ,n , Pseudo − Stabilization : α ∼ α p n , for all α ∈ P M ,n . Note that from the discussion in § L -moves. Hence, we obtained a sharpened version ofTheorem 9. More precisely, we have the following: Theorem 10 ( L -move Markov’s Theorem for mixed pseudo braids). Two mixed pseudobraids have isotopic closures if and only if one can obtained from the other by a finite sequenceof the following moves: L − movesCommuting : α p i ∼ p i α, for all α ∈ P M ,n , Pseudo − Stabilization : α ∼ α p n , for all α ∈ P M ,n . Similarly, by Theorem 7, we have the following result for mixed singular braids:
Theorem 11 ( The analogue of the Markov Theorem for mixed singular braids).
Twomixed singular braids have equivalent closures if and only if one can obtained from the other by finite sequence of the following moves: Commuting : α p i ∼ p i α, for all α ∈ SM ,n , Conjugation : β ∼ α ± β α ∓ for all β ∈ SM ,n & α ∈ B ,n , Real − Stabilization : α ∼ α σ ± n , for all α ∈ SM ,n , Equivalently, by a finite sequence of the following moves: L − movesCommuting : α p i ∼ p i α, for all α ∈ SM ,n , Pseudo Hecke and singular Hecke algebras of type A and type B
In [33] the authors define the singular Hecke algebra of type A, S H n , and through a universalMarkov trace constructed in S H n , they present a HOMFLYPT-type invariant for singular linksin S . In this section we discuss techniques from [25, 33] and [28] toward the construction ofHOMFLYPT-type invariants for pseudo links in S and ST, and for singular links in ST. Theconstruction of these type of invariants will be the subject of a sequel paper.3.1. The pseudo Hecke algebra of type A.
The technique of [33] can be applied for thecase of pseudo knots in S , since, as mentioned before, pseudo knot theory of S can be realizedas the quotient of the theory of singular knots in S , modulo the pseudo-Reidemeister move 1.In that sense, we present here results from [33] which are adapted accordingly.Recall that the Hecke algebra of type A, H n , has presentation obtained from the classicalbraid group B n by corresponding the braiding generators σ i to g i and by adding the quadraticrelations:(2) g i = ( q − g i + q, ≤ i < n, q ∈ C . That is, H n = C [ B n ]
Define the pseudo Hecke algebra of type A , P H n , as the quotient of the pseudobraid monoid C [ P M n ] by the quadratic relations (2). That is, P H n = C [ P M n ] < g i − ( q − g i − q > . Note that the natural embedding
P M n ֒ → P M n +1 induces a homomorphism ι : P H n → P H n +1 . Moreover, P M n has a natural grading with respect to the pre-crossings, i.e. P M n = ∞ ⊕ d =0 P d M n , where P d M n denote the set of mixed pseudo braids with d pre-crossings. This gradinginduces a grading on P H n , since the quadratic relations affect only the g i ’s. In particular, wehave that: P H n = ∞ ⊕ d =0 P d H n . Recall now the following linear basis for the Hecke algebra of type A [25]: S = (cid:8) ( g i g i − . . . g i − k )( g i g i − . . . g i − k ) . . . ( g i p g i p − . . . g i p − k p ) (cid:9) , for 1 ≤ i < . . . < i p ≤ n − he basis S yields directly an inductive basis for H n , which is used in the construction ofthe Ocneanu trace, leading to the HOMFLYPT or 2-variable Jones polynomial. Similarly, wewant to find a basis for P H n . Although P H n is not of finite dimension, each subspace P d H n ofthe graduation is of finite dimension. Indeed we have the following (compare to Proposition 3.1[33]): Proposition 2.
The set C d = { p i p i . . . p i d α, where 1 ≤ i j ≤ n − , ≤ j ≤ d and α ∈ B n } , spans P d H n . The proof is similar to that of Proposition 3.1 [33] by replacing the singular crossings τ i ’s bythe pre-crossings p i ’s.In order now to define HOMFLYPT-type invariants for pseudo links in S , one should con-struct a family of Markov traces on { P d H n } ∞ n =1 and P H n . Note that the difference betweenthe family of traces we have to define in { P d H n } ∞ n =1 to that defined in [33], is the relation thatcomes from the pseudo-Reidemeister move 1, and which is not allowed in the theory of singularknots. In particular, for w ∈ C , we have to impose the following condition:tr( ap n ) = w tr( a ) , for a ∈ P d H n ( cor. P H n ) . In a sequel paper we shall work toward the construction of such Markov traces. Note thatthe construction of a Markov trace leads naturally to a HOMFLYPT-type invariant for pseudolinks in S . As shown in [33], one would have to fix a Markov trace on { P d H} ∞ n =1 , and thenfollow [25] in order to define an invariant X for pseudo knots in S .3.2. On the pseudo Hecke algebras and the singular Hecke algebras of type B.
In thissubsection we define the generalized pseudo Hecke algebra of type B and the cyclotomic pseudoHecke algebras of type B , related to pseudo links in ST, and similarly, we define the cyclotomicand the generalized singular Hecke algebra of type B , related to the theory of singular links in ST,following [28]. Our aim is to apply similar techniques from [28, 33, 25] toward the constructionof HOMFLYPT-type invariants for pseudo links and singular links in the Solid Torus, ST, whichis the subject of a sequel paper.It has been established that Hecke algebras of type B form a tower of B-type algebras thatare related to the knot theory of ST. A presentation for the basic one is obtained from thepresentation of the Artin group B ,n by adding the quadratic relation (2) and t = ( Q − t + Q ,where q, Q ∈ C \{ } are seen as fixed variables. The middle B-type algebras are the cyclotomicHecke algebras of type B, whose presentations are obtained by the quadratic relation (2) and t d = ( t − u )( t − u ) . . . ( t − u d ). The topmost Hecke-like algebra in the tower is the generalizedHecke algebra of type B , H ,n , which has the following presentation:H ,n ( q ) = * t, g , . . . , g n − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) g tg t = tg tg tg i = g i t, i > g i g i +1 g i = g i +1 g i g i +1 , ≤ i ≤ n − g i g j = g j g i , | i − j | > g i = ( q − g i + q, i = 1 , . . . , n − + . That is: H ,n ( q ) = Z (cid:2) q ± (cid:3) B ,n h σ i − ( q − σ i − q i . Note that in H ,n ( q ) the generator t satisfies no polynomial relation, making the algebraH ,n ( q ) infinite dimensional. Also that in [28] the algebra H ,n ( q ) is denoted as H n ( q, ∞ ). ollowing [28, 33], we now define the generalized pseudo Hecke algebra of type B and thecyclotomic pseudo Hecke algebras of type B. Definition 8.
Define the generalized pseudo Hecke algebra of type B , P H ,n , as the quotient ofthe mixed pseudo braid monoid C [ P M ,n ] by the quadratic relations (2). That is, P H ,n = C [ P M ,n ] < g i − ( q − g i − q > . The cyclotomic pseudo Hecke algebras of type B, P H c ,n is defined as the quotient of themixed pseudo braid monoid C [ P M ,n ] by the quadratic relations (2) and the relations: t c = ( t − u c )( t − u c ) . . . ( t − u c ) . As in the case of pseudo Hecke algebras of type A, the natural embedding
P M ,n ֒ → P M ,n +1 induces a homomorphism ι : P H ,n → P H ,n +1 , and moreover, P M ,n also has a natural gradingwith respect to the pre-crossings, i.e. P M ,n = ∞ ⊕ d =0 P d M ,n , where P d M ,n denote the set ofmixed pseudo braids with d pre-crossings. This grading induces a grading on P H ,n : P H ,n = ∞ ⊕ d =0 P d H ,n . Consider now the elements illustrated in Figure 15 and recall the following linear bases forthe generalized Hecke algebra of type B [28]:( i ) Σ n = { t k i . . . t k r i r · σ } , where 0 ≤ i < . . . < i r ≤ n − , ( ii ) Σ ′ n = { t ′ i k . . . t ′ i r k r · σ } , where 0 ≤ i < . . . < i r ≤ n − , where k , . . . , k r ∈ Z and σ a basic element in H n ( q ). Figure 15.
The elements t ′ i and t i .The basis Σ ′ n yields an inductive basis for the generalized Hecke algebra of type B, which isused in the construction of the Ocneanu trace, leading to the HOMFLYPT or 2-variable Jonespolynomial (the reader is referred to [28] for more details). Similarly, we want to find a basis for P H ,n , with the use of which we will construct HOMFLYPT-type invariants for pseudo links inST. This is more complicated (compared to the type A case) and further research is required.We conclude this discussion with a conjecture: Conjecture 1.
For σ ∈ P d H n , the following sets span P d H ,n : ( i ) C dn = { t k i . . . t k r i r · σ } , where 0 ≤ i < . . . < i r ≤ n − , ( ii ) C ′ dn = { t ′ i k . . . t ′ i r k r · σ } , where 0 ≤ i < . . . < i r ≤ n − . emark 4. For singular links in ST, we define the generalized singular Hecke algebra of typeB , S H ,n , as S H ,n = C [ SM ,n ] < g i − ( q − g i − q > , and the cyclotomic singular Hecke algebras of type B , S H c ,n , as the following quotient: S H c ,n = C [ SM ,n ] < g i − ( q − g i − q, t c = ( t − u c )( t − u c ) . . . ( t − u c ) > . Similarly to Conjecture 1, the sets( i ) C dn = { t k i . . . t k r i r · σ } , where 0 ≤ i < . . . < i r ≤ n − , ( ii ) C ′ dn = { t ′ i k . . . t ′ i r k r · σ } , where 0 ≤ i < . . . < i r ≤ n − . , where σ ∈ S d H n , are potential spanning sets of S d H ,n . Remark 5.
It is worth mentioning that, as shown in [16, 17, 18, 12], the generalized Heckealgebra of type B is related to the knot theory of lens spaces L ( p, q ). This can be realized bythe fact that one may generalize a HOMFLYPT-type invariant for links in ST to an invariantfor links in L ( p, q ) by imposing relations coming from the braid band moves , that is, moves thatreflect isotopy in L ( p, q ) and which are similar to the second Kirby move [30, 14].4. The pseudo bracket polynomial for pseudo links in ST
In this section we recall results from [31, 24] on the pseudo bracket polynomial for pseudolinks in S and we extend this polynomial for the case of pseudo links in ST.4.1. The pseudo bracket polynomial for pseudo links in S . In [24] the pseudo bracketpolynomial, < ; > , is defined for pseudo links in S extending the Kauffman bracket polynomialfor classical knots presented in [31]. Note that the orientation of a diagram in the case of pseudolinks is needed in order to define a skein relation on pre-crossings. More precisely, we have thefollowing: Definition 9.
Let L be an oriented pseudo link in S . The pseudo bracket polynomial of L isdefined by means of the following relations: t can be easily seen that the pseudo bracket is invariant under Reidemeister moves 2 and 3 andthe pseudo moves PR1, PR2 and PR3 ([24] Theorem 1). As in [31], by normalizing the pseudobracket polynomial using the writhe, we obtain the normalized pseudo bracket polynomial , whichis an invariant of pseudo knots in S ([24], Corollary 2). In particular: Theorem 12.
Let K be a pseudo diagram of a pseudo knot. The polynomial P K ( A, V ) = ( − A ) w ( K ) < K >, where w ( K ) := P c ∈ C ( K ) sgn ( c ) , C ( k ) the set of classical crossings of K and < K > the pseudobracket polynomial of K , is an invariant of pseudo knots in S . Remark 6.
For the case of singular links in S the reader is referred to [7], where three differentapproaches to the bracket polynomial for singular links in S are presented.4.2. The pseudo bracket polynomial for pseudo knots in ST.
We now generalize thepseudo bracket polynomial for pseudo links in ST. Note that we now view ST as a puncturedtorus (for an illustration of the looping generator t in this set up see Figure 16). Figure 16.
The looping generator t in the new set up.We have the following: Definition 10.
Let L be an oriented pseudo link in ST. The pseudo bracket polynomial of L isdefined by means of the relations in Definition 9 together with the following relations:An immediate result of Theorem 1 [24] is the following: Proposition 3.
The pseudo bracket polynomial for pseudo knots in ST is invariant underReidemeister moves 2 and 3 and all pseudo moves.
As in the case of pseudo links in S we obtain the following result: Corollary 2.
Let K be a pseudo diagram of a pseudo knot in ST. The polynomial P K ( A, V, s ) = ( − A ) w ( K ) < K >, where w ( K ) is the writhe of the pseudo knots and < K > the pseudo bracket polynomial of K ,is an invariant of pseudo knots in ST. emark 7. i. For the case of the bracket polynomial for singular knots in ST the readeris referred to [5].ii. It is well known that the Temperley-Lieb algebras are related to the Kauffman bracketpolynomial (see [32]). In [10], and with the use of the Tempereley–Lieb algebra oftype B, an alternative basis for the Kauffman bracket skein module of the solid torusis presented. It would be interesting to construct Kauffman-type invariants for pseudolinks in ST using a generalized pseudo Temperley-Lieb algebra, and then extend thisinvariant to and invariant for pseudo links in the lens spaces L ( p, q ).5. Conclusions
In this paper we introduce and study pseudo links in ST via the mixed pseudo braid monoid. Inparticular, we present the appropriate topological set up toward the construction of HOMFLYPT-type invariants for pseudo links in S and in ST. We believe that pseudo knots can be used invarious aspects of molecular biology, and in particular, they could serve as a model to biologicalobjects related to DNA. We also introduce and study singular links in ST and we relate thesingular knot theory of ST to that of pseudo knot theory of ST. In a sequel paper we shall adoptthese results in order to construct HOMFLYPT-type invariants of pseudo links in S and ST(cor. for singular links in ST) via the generalized pseudo Hecke algebra of type B (cor. thegeneralized singular Hecke algebras of type B). Finally, it is worth mentioning that in [13] wework toward formulating the analogues of the Alexander and the Markov theorems for pseudolinks in various 3-manifolds. References [1]
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China Agricultural University, International College Beijing, No.17 Qinghua East Road, Haid-ian District, Beijing, 100083, P. R. China.
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