The braid approach to the HOMFLYPT skein module of the lens spaces L(p,1) (original) (raw)
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TOPOLOGICAL STEPS TOWARD THE HOMFLYPT SKEIN MODULE OF THE LENS SPACES L(p, 1) VIA BRAIDS
In this paper we work toward the Homflypt skein module of the lens spaces L(p, 1), S(L(p, 1)), using braids. In particular, we establish the connection between S(ST), the Homflypt skein module of the solid torus ST, and S(L(p, 1)) and arrive at an infinite system, whose solution corresponds to the computation of S(L(p, 1)). We start from the Lambropoulou invariant X for knots and links in ST, the universal analogue of the Homflypt polynomial in ST, and a new basis, Λ, of S(ST) presented in [DL1]. We show that S(L(p, 1)) is obtained from S(ST) by considering relations coming from the performance of braid band moves (bbm) on elements in the basis Λ, where the braid band moves are performed on any moving strand of each element in Λ. We do that by proving that the system of equations obtained from diagrams in ST by performing bbm on any moving strand is equivalent to the system obtained if we only consider elements in the basic set Λ. The importance of our approach is that it can shed light to the problem of computing skein modules of arbitrary c.c.o. 3-manifolds, since any 3-manifold can be obtained by surgery on S^3 along unknotted closed curves. The main difficulty of the problem lies in selecting from the infinitum of band moves some basic ones and solving the infinite system of equations.
Journal of Knot Theory and Its Ramifications
We prove that, in order to derive the HOMFLYPT skein module of the lens spaces [Formula: see text] from the HOMFLYPT skein module of the solid torus, [Formula: see text], it suffices to solve an infinite system of equations obtained by imposing on the Lambropoulou invariant [Formula: see text] for knots and links in the solid torus, braid band moves that are performed only on the first moving strand of elements in a set [Formula: see text], augmenting the basis [Formula: see text] of [Formula: see text].
HOMFLYPT SKEIN SUB-MODULES OF THE LENS SPACES L(p, 1)
2021
In this paper we work toward the HOMFLYPT skein module of L(p, 1), S(L(p, 1)), via braids. Our starting point is the linear Turaev-basis, Λ ′ , of the HOMFLYPT skein module of the solid torus ST, S(ST), which can be decomposed as the tensor product of the "positive" Λ ′+ and the "negative" Λ ′− sub-modules, and the Lambropoulou invariant, X, for knots and links in ST, that captures S(ST). It is a well-known result by now that S(L(p, 1)) = S(ST) <bbm ′ s> , where bbm's (braid band moves) denotes the isotopy moves that correspond to the surgery description of L(p, 1). Namely, a HOMFLYPT-type invariant for knots and links in ST can be extended to an invariant for knots and links in L(p, 1) by imposing relations coming from the performance of bbm's and solving the infinite system of equations obtained that way. In this paper we work with a new basis of S(ST), Λ, and we relate the infinite system of equations obtained by performing bbm's on elements in Λ + to the infinite system of equations obtained by performing bbm's on elements in Λ − via a map I. More precisely we prove that the solutions of one system can be derived from the solutions of the other. Our aim is to reduce the complexity of the infinite system one needs to solve in order to compute S(L(p, 1)) using the braid technique. Finally, we present a generating set and a potential basis for Λ + <bbm ′ s> and thus, we obtain a generating set and a potential basis for Λ − <bbm ′ s>. We also discuss further steps needed in order to compute S(L(p, 1)) via braids.
A new basis for the Homflypt skein module of the solid torus
In this paper we give a new basis, Λ, for the Homflypt skein module of the solid torus, S(ST), which topologically is compatible with the handle sliding moves and which was predicted by J.H. Przytycki. The basis Λ is different from the basis Λ , discovered independently by Hoste and Kidwell [1] and Turaev [2] with the use of diagrammatic methods, and also different from the basis of Morton and Aiston [3]. For finding the basis Λ we use the generalized Hecke algebra of type B, H_{1,n} , which is generated by looping elements and braiding elements and which is isomorphic to the affine Hecke algebra of type A [4]. More precisely, we start with the well-known basis Λ of S(ST) and an appropriate linear basis Σ_n of the algebra H_{1,n}. We then convert elements in Λ to sums of elements in Σ_n. Then, using conjugation and the stabilization moves, we convert these elements to sums of elements in Λ by managing gaps in the indices, by ordering the exponents of the looping elements and by eliminating braiding tails in the words. Further, we define total orderings on the sets Λ and Λ and, using these orderings, we relate the two sets via a block diagonal matrix, where each block is an infinite lower triangular matrix with invertible elements in the diagonal. Using this matrix we prove linear independence of the set Λ, thus Λ is a basis for S(ST). S(ST) plays an important role in the study of Homflypt skein modules of arbitrary c.c.o. 3-manifolds, since every c.c.o. 3-manifold can be obtained by integral surgery along a framed link in S 3 with unknotted components. In particular, the new basis of S(ST) is appropriate for computing the Homflypt skein module of the lens spaces. In this paper we provide some basic algebraic tools for computing skein modules of c.c.o. 3-manifolds via algebraic means.
The HOMFLY polynomial of links in closed braid form
Discrete Mathematics
It is well known that any link can be represented by the closure of a braid. The minimum number of strings needed in a braid whose closure represents a given link is called the braid index of the link and the well known Morton-Frank-Williams inequality reveals a close relationship between the HOMFLY polynomial of a link and its braid index. In the case that a link is already presented in a closed braid form, Jaeger derived a special formulation of the HOMFLY polynomial. In this paper, we prove a variant of Jaeger's result as well as a dual version of it. Unlike Jaeger's original reasoning, which relies on representation theory, our proof uses only elementary geometric and combinatorial observations. Using our variant and its dual version, we provide a direct and elementary proof of the fact that the braid index of a link that has an n-string closed braid diagram that is also reduced and alternating, is exactly n. Until know this fact was only known as a consequence of a result due to Murasugi on fibered links that are star products of elementary torus links and of the fact that alternating braids are fibered.
Knots, Low-Dimensional Topology and Applications
We summarize the theory of a new skein invariant of classical links H [H ] that generalizes the regular isotopy version of the Homflypt polynomial, H. The invariant H [H ] is based on a procedure where we apply the skein relation only to crossings of distinct components, so as to produce collections of unlinked knots and then we evaluate the resulting knots using the invariant H and inserting at the same time a new parameter. This procedure, remarkably, leads to a generalization of H but also to generalizations of other known skein invariants, such as the Kauffman polynomial. We discuss the different approaches to the link invariant H [H ], the algebraic one related to its ambient isotopy equivalent invariant , the skein-theoretic one and its reformulation into a summation of the generating invariant H on sublinks of a given link. We finally give examples illustrating the behaviour of the invariant H [H ] and we discuss further research directions and possible application areas. Keywords Classical links • Yokonuma-Hecke algebras • Mixed crossings • Reidemeister moves • Stacks of knots • Homflypt polynomial • Kauffman polynomial • Dubrovnik polynomial • Skein relations • Skein invariants • 3-variable link invariant • Closed combinatorial formulae
International Journal of Modern Physics A, 2012
We continue the program of systematic study of extended HOMFLY polynomials, suggested in [A. Mironov, A. Morozov and And. Morozov, arXiv:1112.5754] and [A. Mironov, A. Morozov and And. Morozov, J. High Energy Phys. 03, 034 (2012), arXiv:1112.2654]. Extended polynomials depend on infinitely many time-variables, are close relatives of integrable τ-functions, and depend on the choice of the braid representation of the knot. They possess natural character decompositions, with coefficients which can be defined by exhaustively general formula for any particular number m of strands in the braid and any particular representation R of the Lie algebra GL(∞). Being restricted to "the topological locus" in the space of time-variables, the extended HOMFLY polynomials reproduce the ordinary knot invariants. We derive such a general formula, for m = 3, when the braid is parametrized by a sequence of integers (a1, b1, a2, b2, …) and for the first nonfundamental representation R = [2]. Ins...
Polynomial invariants of knots in lens spaces
Topology and its Applications, 1991
We characterize the polynomials that can arise as Alexander polynomial of (homotopically nontrivial) knots in classical lens spaces. Our main geometric argument contains an algorithm to produce knots realizing these polynomials.
The Quantum sl<sub>2</sub>-Invariant of a Family of Knots
Applied Mathematics, 2014
We give a general formula of the quantum 2 sl-invariant of a family of braid knots. To compute the quantum invariant of the links we use the Lie algebra 2 g sl = in its standard two-dimensional representation. We also recover the Jones polynomial of these knots as a special case of this quantum invariant.
Journal of Knot Theory and Its Ramifications
We study new skein invariants of links based on a procedure where we first apply a given skein relation only to crossings of distinct components, so as to produce collections of unlinked knots. We then evaluate the resulting knots using the given invariant. A skein invariant can be computed on each link solely by the use of skein relations and a set of initial conditions. The new procedure, remarkably, leads to generalizations of the known skein invariants. We make skein invariants of classical links, [Formula: see text], [Formula: see text] and [Formula: see text], based on the invariants of knots, [Formula: see text], [Formula: see text] and [Formula: see text], denoting the regular isotopy version of the Homflypt polynomial, the Kauffman polynomial and the Dubrovnik polynomial. We provide skein theoretic proofs of the well-definedness of these invariants. These invariants are also reformulated into summations of the generating invariants ([Formula: see text], [Formula: see text],...