Narrow Results

Given a knot theory (virtual, singular, knots in a 3-manifold etc.), there are deep relations between the diagrammatic knot equivalence in this theory, the braid structures and a corresponding braid equivalence. The L-moves between braids, due to their fundamental nature, may be adapted to any diagrammatic situation in order to formulate a corresponding braid equivalence. In this short paper, we discuss and compare various diagrammatic set-ups and results therein, in order to draw the underlying logic relating diagrammatic isotopy, braid structures, Markov theorems and L-move analogues. Finally, we apply our conclusions to singular 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 ${𝒮}(\mathrm{\text{ST}})$, 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 analog of the Homflypt polynomial in ST, and a new basis, $\mathrm{\Lambda }$, of ${𝒮}(\mathrm{\text{ST}})$ presented in [I. Diamantis and S. Lambropoulou, A new basis for the Homflypt skein module of the solid torus, J. Pure Appl. Algebra220(2) (2016) 577–605, http://dx.doi.org/10.1016/j.jpaa.2015.06.014, arXiv:1412.3642 [math.GT]]. We show that ${𝒮}(L(p,1))$ is obtained from ${𝒮}(\mathrm{\text{ST}})$ by considering relations coming from the performance of braid band move(s) [bbm] on elements in the basis $\mathrm{\Lambda }$, where the bbm are performed on any moving strand of each element in $\mathrm{\Lambda }$. 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 $\mathrm{\Lambda }$. The importance of our approach is that it can shed light on 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.

We prove that, in order to derive the HOMFLYPT skein module of the lens spaces $L(p,1)$ from the HOMFLYPT skein module of the solid torus, ${𝒮}(\mathrm{\text{ST}})$, it suffices to solve an infinite system of equations obtained by imposing on the Lambropoulou invariant $X$ 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 ${\mathrm{\Lambda }}^{\mathrm{\text{aug}}}$, augmenting the basis $\mathrm{\Lambda }$ of ${𝒮}(\mathrm{\text{ST}})$.

In this paper we present two new bases, $B_{H_2}^{\prime }$ and ${{ℬ}}_{H_2}$, for the Kauffman bracket skein module of the handlebody of genus 2 $H_2$, KBSM$\left(H_2\right)$. We start from the well-known Przytycki-basis of KBSM$\left(H_2\right)$, $B_{H_2}$, and using the technique of parting we present elements in $B_{H_2}$ in open braid form. We define an ordering relation on an augmented set $L$ consisting of monomials of all different “loopings” in $H_2$, that contains the sets $B_{H_2}$, $B_{H_2}^{\prime }$ and ${{ℬ}}_{H_2}$ as proper subsets. Using the Kauffman bracket skein relation we relate $B_{H_2}$ to the sets $B_{H_2}^{\prime }$ and ${{ℬ}}_{H_2}$ via a lower triangular infinite matrix with invertible elements in the diagonal. The basis $B_{H_2}^{\prime }$ is an intermediate step in order to reach at elements in ${{ℬ}}_{H_2}$ that have no crossings on the level of braids, and in that sense, ${{ℬ}}_{H_2}$ is a more natural basis of KBSM$\left(H_2\right)$. Moreover, this basis is appropriate in order to compute Kauffman bracket skein modules of closed–connected–oriented (c.c.o.) 3-manifolds $M$ that are obtained from $H_2$ by surgery, since isotopy moves in $M$ are naturally described by elements in ${{ℬ}}_{H_2}$.

Tied links in $S^3$ were introduced by Aicardi and Juyumaya as standard links in $S^3$ equipped with some non-embedded arcs, called ties, joining some components of the link. Tied links in the Solid Torus were then naturally generalized by Flores. In this paper, we study this new class of links in other topological settings. More precisely, we study tied links in the lens spaces $L(p,1)$, in handlebodies of genus $g$, and in the complement of the $g$-component unlink. We introduce the tied braid monoids $T{M}_{g,n}$ by combining the algebraic mixed braid groups defined by Lambropoulou and the tied braid monoid, and we formulate and prove analogues of the Alexander and the Markov theorems for tied links in the 3-manifolds mentioned above. We also present an $L$-move braid equivalence for tied braids and we discuss further research related to tied links in knot complements and c.c.o. 3-manifolds. The theory of tied links has potential use in some aspects of molecular biology.

In this paper, we develop a braid theoretic approach for computing the Kauffman bracket skein module of the lens spaces $L(p,q)$, KBSM($L(p,q)$), for $q\ne 0$. For doing this, we introduce a new concept, that of an unoriented braid. Unoriented braids are obtained from standard braids by ignoring the natural top-to-bottom orientation of the strands. We first define the generalized Temperley–Lieb algebra of type B, ${\mathrm{\text{TL}}}_{1,n}$, which is related to the knot theory of the solid torus ST, and we obtain the universal Kauffman bracket-type invariant, $V$, for knots and links in ST, via a unique Markov trace constructed on ${\mathrm{\text{TL}}}_{1,n}$. The universal invariant $V$ is equivalent to the KBSM(ST). For passing now to the KBSM($L(p,q)$), we impose on $V$ relations coming from the band moves (or slide moves), that is, moves that reflect isotopy in $L(p,q)$ but not in ST, and which reflect the surgery description of $L(p,q)$, obtaining thus, an infinite system of equations. By construction, solving this infinite system of equations is equivalent to computing KBSM($L(p,q)$). We first present the solution for the case $q=1$, which corresponds to obtaining a new basis, ${{ℬ}}_p$, for KBSM($L(p,1)$) with $(\lfloor p/2\rfloor +1)$ elements. We note that the basis ${{ℬ}}_p$ is different from the one obtained by Hoste and Przytycki. For dealing with the complexity of the infinite system for the case $q>1$, we first show how the new basis ${{ℬ}}_p$ of KBSM($L(p,1)$) can be obtained using a diagrammatic approach based on unoriented braids, and we finally extend our result to the case $q>1$. The advantage of the braid theoretic approach that we propose for computing skein modules of c.c.o. 3-manifolds, is that the use of braids provides more control on the isotopies of knots and links in the manifolds, and much of the diagrammatic complexity is absorbed into the proofs of the algebraic statements.