Ioannis Diamantis | China Agricultural University (original) (raw)

Uploads

Papers by Ioannis Diamantis

In this paper we give an alternative basis, BST, for the Kauffman bracket skein module of the sol... more In this paper we give an alternative basis, BST, for the Kauffman bracket skein module of the solid torus, KBSM (ST). The basis BST is obtained with the use of the Tempereley-Lieb algebra of type B and it is appropriate for computing the Kauffman bracket skein module of the lens spaces L(p, q) via braids.

We prove that, in order to derive the HOMFLYPT skein module of the lens spaces L(p, 1) from the H... more 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(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 Λ aug , augmenting the basis Λ of S(ST).

In this paper we present recent results toward the computation of the HOMFLYPT skein module of th... more In this paper we present recent results toward the computation of the HOMFLYPT skein module of the lens spaces L(p,1)L(p,1)L(p,1), mathcalSleft(L(p,1)right)\mathcal{S}\left(L(p,1) \right)mathcalSleft(L(p,1)right), via braids. Our starting point is the knot theory of the solid torus ST and the Lambropoulou invariant, XXX, for knots and links in ST, the universal analogue of the HOMFLYPT polynomial in ST. The relation between mathcalSleft(L(p,1)right)\mathcal{S}\left(L(p,1) \right)mathcalSleft(L(p,1)right) and mathcalS(rmST)\mathcal{S}({\rm ST})mathcalS(rmST) is established in \cite{DLP} and it is shown that in order to compute mathcalSleft(L(p,1)right)\mathcal{S}\left(L(p,1) \right)mathcalSleft(L(p,1)right), it suffices to solve an infinite system of equations obtained by performing all possible braid band moves on elements in the basis of mathcalS(rmST)\mathcal{S}({\rm ST})mathcalS(rmST), Lambda\LambdaLambda, presented in \cite{DL2}. The solution of this infinite system of equations is very technical and is the subject of a sequel paper \cite{DL3}.

In this paper we work toward the Homflypt skein module of the lens spaces L(p, 1), S(L(p, 1)), us... more 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.

In this paper we give a new basis, Λ, for the Homflypt skein module of the solid torus, S(ST), wh... more 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.

In this paper we provide algebraic mixed braid classification of links in any c.c.o. 3-manifold M... more In this paper we provide algebraic mixed braid classification of links in any c.c.o. 3-manifold M obtained by rational surgery along a framed link in S^3. We do this by representing M by a closed framed braid in S^3 and links in M by closed mixed braids in S^3. We first prove an analogue of the Reidemeister theorem for links in M. We then give geometric formulations of the mixed braid equivalence using the L-moves and the braid band moves. Finally we formulate the algebraic braid equivalence in terms of the mixed braid groups B_{m,n}, using cabling and the parting and combing techniques for mixed braids. Our results set a homogeneous algebraic ground for studying links in 3-manifolds and in families of 3-manifolds using computational tools. We provide concrete formuli of the braid equivalences in lens spaces, in Seifert manifolds, in homology spheres obtained from the trefoil and in manifolds obtained from torus knots.
Our setting is appropriate for constructing Jones-type invariants for links in families of 3-manifolds via quotient algebras of the mixed braid groups B_{m,n}, as well as for studying skein modules of 3-manifolds, since they provide a controlled algebraic framework and much of the diagrammatic complexity has been absorbed into the proofs. Further, our moves can be used in a braid analogue of Rolfsen’s rational calculus and potentially in computing Witten invariants.