27 pages, with about 20 embedded figures. Two misprints in the polynomials for the trefoil and figure eight knots in the last paragraph are corrected. We are grateful to Marcos Marino for noticing one of the errors
The meridian maps of the full Homfly skein of the annulus are linear endomorphisms induced by the insertion of a meridian loop, with either orientation, around a diagram in the annulus. The eigenvalues of the meridian maps are known to be distinct, and are indexed by pairs of partitions of integers p and n into k and k* parts respectively.
We give here an explicit formula for a corresponding eigenvector as the determinant of a (k*+k)x(k*+k) matrix whose entries are skein elements corresponding to partitions with a single part. This extends the results of Kawagoe and Lukac, for the case p=0, giving a basis for the subspace of the skein spanned by closed braids all oriented in the same direction. Their formula uses the Jacobi-Trudy determinants for Schur functions in terms of complete symmetric functions. Our matrices have a similar pattern of entries in k rows, and a modified pattern in k* rows, resulting in a combination of closed braids with up to n strings oriented in one direction and p in the reverse direction.
We discuss the 2-variable knot invariants resulting from decoration of a knot by these skein elements, and their relation to unitary quantum invariants.
For a ring $R$, we denote by $R[\mathcal L]$ the free $R$-module spanned by the isotopy classes of singular links in $\mathbb S^3$. Given two invertible elements $x,t \in R$, the HOMFLY-PT skein module of singular links in $\mathbb S^3$ (relative to the triple $(R,t,x)$) is the quotient of $R[\mathcal L]$ by local relations, called skein relations, that involve $t$ and $x$. We compute the HOMFLY-PT skein module of singular links for any $R$ such that $(t^{-1}-t+x)$ and $(t^{-1}-t-x)$ are invertible. In particular, we deduce the Conway skein module of singular links.
We introduce a formula for the action of Dehn twists on the HOMFLY-PT type skein module of a surface. As an application of the formula to mapping class group, we give an embedding from the Torelli group of a surface $Σ_{g,1}$ of genus $g$ with non-empty connected boundary into the completed HOMFLY-PT type skein algebra. As an application of the formula to integral homology $3$-spheres, we construct an invariant $z(M) \in \mathbb{Q} [ρ] [[h]]$ for an integral homology $3$-sphere $M$. The invariant $z(M) \mod (h^{n+1})$ is a finite type invariant of order $n$.
A homology cylinder of a surface induces an automorphism of the completed group ring of the fundamental group of the surface. We introduce a new method of computing the automorphism by using the Goldman Lie algebra of the surface or some skein algebra. In particular, we give a refinement of a formula by Kuno and Massuyeau.
The oriented framed Homfly skein C of the annulus provides the natural parameter space for the Homfly satellite invariants of a knot. It contains a submodule C+ isomorphic to the algebra of the symmetric functions. We collect and expand formulae relating elements expressed in terms of symmetric functions to Turaev's geometrical basis of C+. We reformulate the formulae of Rosso and Jones for quantum sl(N) invariants of cables in terms of plethysms of symmetric functions, and use the connection between quantum sl(N) invariants and C+ to give a formula for the satellite of a cable as an element o…
We work in the reduced SU(N,K) modular category as constructed recently by Blanchet. We define spin type and cohomological refinements of the Turaev-Viro invariants of closed oriented 3-manifolds and give a formula relating them to Blanchet's invariants. Roberts' definition of the Turaev-Viro state sum is exploited. Furthermore, we construct refined Turaev-Viro and Reshetikhin-Turaev TQFT's and study the relationship between them.
In this paper, we study the properties of the colored HOMFLY polynomials via HOMFLY skein theory. We prove some limit behaviors and symmetries of the colored HOMFLY polynomial predicted in some physicists' recent works.
In \cite{GZ}, Gilmer and Zhong established the existence of an invariant for links in $S^1\times S^2$ which is a rational function in variables $a$ and $s$ and satisfies the HOMFLY-PT skein relations. We give formulas for evaluating this invariant in terms of a standard, geometrically simple basis for the HOMFLY-PT skein module of the solid torus. This allows computation of the invariant for arbitrary links in $S^1\times S^2$ and shows that the invariant is in fact a Laurent polynomial in $a$ and $z= s -s^{-1}$. Our proof uses connections between HOMFLY-PT skein modules and invariants of Legen…
By using the HOMFLY skein theory. We prove a strong integrality theorem for the reduced colored HOMFLYPT invariants defined by a basis in the full HOMFLY skein of the annulus.
In this paper, we investigate the properties of the full colored HOMFLYPT invariants in the full skein of the annulus $\mathcal{C}$. We show that the full colored HOMFLYPT invariant has a nice structure when $q\rightarrow 1$. The composite invariant is a combination of the full colored HOMFLYPT invariants. In order to study the framed LMOV type conjecture for composite invariants, we introduce the framed reformulated composite invariant $\check{\mathcal{R}}_{p}(\mathcal{L})$. By using the HOMFLY skein theory, we prove that $\check{\mathcal{R}}_{p}(\mathcal{L})$ lies in the ring $2\mathbb{Z}[(q…
In this paper we work toward the HOMFLYPT skein module of $L(p, 1)$, $\mathcal{S}(L(p,1))$, via braids. Our starting point is the linear Turaev-basis, $Λ^{\prime}$, of the HOMFLYPT skein module of the solid torus ST, $\mathcal{S}({\rm ST})$, which can be decomposed as the tensor product of the "positive" ${Λ^{\prime}}^+$ and the "negative" ${Λ^{\prime}}^-$ sub-modules, and the Lambropoulou invariant, $X$, for knots and links in ST, that captures $S({\rm ST})$. It is a well-known result by now that $\mathcal{S}(L(p, 1))=\frac{\mathcal{S}(ST)}{<bbm's>}$, where bbm's (braid band moves) denotes th…
Colored HOMFLY-PT invariant, the generalization of the colored Jones polynomial, is one of the most important quantum invariants of links. This paper is devoted to investigating the basic structures of the colored HOMFLY-PT invariants of links. By using the HOMFLY-PT skein theory, firstly, we show that the (reformulated) colored HOMFLY-PT invariants actually lie in the ring $\mathbb{Z}[(q-q^{-1})^2,t^{\pm 1}]$. Secondly, we establish some symmetric formulas for colored HOMFLY-PT invariants of links, which include the rank-level duality as an easy consequence. Finally, motivated by the Labastid…
The Murphy operators in the Hecke algebra H_n of type A are explicit commuting elements, whose symmetric functions are central in H_n. In [Skein theory and the Murphy operators, J. Knot Theory Ramif. 11 (2002), 475-492] I defined geometrically a homomorphism from the Homfly skein C of the annulus to the centre of each algebra H_n, and found an element P_m in C, independent of n, whose image, up to an explicit linear combination with the identity of H_n, is the m-th power sum of the Murphy operators. The aim of this paper is to give simple geometric representatives for the elements P_m, and to …
The Hecke algebra H_n contains well known idempotents E_λ which are indexed by Young diagrams with n cells. They were originally described by Gyoja. A skein theoretical description of E_λ was given by Aiston and Morton. The closure of E_λ becomes an element Q_λ of the skein of the annulus. In this skein, they are known to obey the same multiplication rule as the symmetric Schur functions s_λ. But previous proofs of this fact used results about quantum groups which were far beyond the scope of skein theory. Our elementary proof uses only skein theory and basic algebra.
In this paper we give a new basis, $Λ$, for the Homflypt skein module of the solid torus, $\mathcal{S}({\rm ST})$, which was predicted by Jozef Przytycki, using topological interpretation. The basis $Λ$ is different from the basis $Λ^{\prime}$, discovered independently by Hoste--Kidwell \cite{HK} and Turaev \cite{Tu} with the use of diagrammatic methods. For finding the basis $Λ$ we use the generalized Hecke algebra of type B, $\textrm{H}_{1,n}$, defined by the second author in \cite{La2}, which is generated by looping elements and braiding elements and which is isomorphic to the affine Hecke …
Let $k$ be a subring of the field of rational functions in $x, v, s$ which contains $x^{\pm 1}, v^{\pm 1}, s^{\pm 1}$. If $M$ is an oriented 3-manifold, let $S(M)$ denote the Homflypt skein module of $M$ over $k$. This is the free $k$-module generated by isotopy classes of framed oriented links in $M$ quotiented by the Homflypt skein relations: (1) $x^{-1}L_{+}-xL_{-}=(s-s^{-1})L_{0}$; (2) $L$ with a positive twist $=(xv^{-1})L$; (3) $L\sqcup O=(\frac{v-v^{-1}}{s-s^{-1}})L$ where $O$ is the unknot. We give two bases for the relative Homflypt skein module of the solid torus with 2 points in the…
We define composite DAHA-superpolynomials of torus knots, depending on pairs of Young diagrams and generalizing the composite HOMFLY-PT polynomials in the theory of the skein of the annulus. We provide various examples. Our superpolynomials extend the DAHA-Jones (refined) polynomials and satisfy all standard symmetries of the DAHA-superpolynomials of torus knots. The latter are conjecturally related to the HOMFLY-PT homology; such a connection is a challenge in the theory of the annulus. At the end, we construct two DAHA-hyperpolynomials extending the DAHA-Jones polynomials of type E and close…
