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Morrison, Walker, and Wedrich used the blob complex to construct a generalization of Khovanov-Rozansky homology to links in the boundary of a 4-manifold. The degree zero part of their theory, called the skein lasagna module, admits an elementary definition in terms of certain diagrams in the 4-manifold. We give a description of the skein lasagna module for 4-manifolds without 1- and 3-handles, and present some explicit calculations for disk bundles over $S^2$.
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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) $Lsqcup O=(frac{v-v^{-1}}{s-s^{-1}})L$ where $O$ is the unknot. We give two bases for the relative Homflypt skein module of the solid torus with 2 points in the boundary. The first basis is related to the basis of $S(S^1times D^2)$ given by V. Turaev and also J. Hoste and M. Kidwell; the second basis is related to a Young idempotent basis for $S(S^1times D^2)$ based on the work of A. Aiston, H. Morton and C. Blanchet. We prove that if the elements $s^{2n}-1$, for $n$ a nonzero integer, and the elements $s^{2m}-v^{2}$, for any integer $m$, are invertible in $k$, then $S(S^{1} times S^2)=k$-torsion module $oplus k$. Here the free part is generated by the empty link $phi$. In addition, if the elements $s^{2m}-v^{4}$, for $m$ an integer, are invertible in $k$, then $S(S^{1} times S^2)$ has no torsion. We also obtain some results for more general $k$.
The Hecke algebra H_n contains well known idempotents E_{lambda} which are indexed by Young diagrams with n cells. They were originally described by Gyoja. A skein theoretical description of E_{lambda} was given by Aiston and Morton. The closure of E_{lambda} becomes an element Q_{lambda} of the skein of the annulus. In this skein, they are known to obey the same multiplication rule as the symmetric Schur functions s_{lambda}. 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.
This paper is a brief overview of some of our recent results in collaboration with other authors. The cocycle invariants of classical knots and knotted surfaces are summarized, and some applications are presented.
An algebraic system is proposed that represent surface cobordisms in thickened surfaces. Module and comodule structures over Frobenius algebras are used for representing essential curves. The proposed structure gives a unified algebraic view of states of categorified Jones polynomials in thickened surfaces and virtual knots. Constructions of such system are presented.
Three new knot invariants are defined using cocycles of the generalized quandle homology theory that was proposed by Andruskiewitsch and Gra~na. We specialize that theory to the case when there is a group action on the coefficients. First, quandle modules are used to generalize Burau representations and Alexander modules for classical knots. Second, 2-cocycles valued in non-abelian groups are used in a way similar to Hopf algebra invariants of classical knots. These invariants are shown to be of quantum type. Third, cocycles with group actions on coefficient groups are used to define quandle cocycle invariants for both classical knots and knotted surfaces. Concrete computational methods are provided and used to prove non-invertibility for a large family of knotted surfaces. In the classical case, the invariant can detect the chirality of 3-colorable knots in a number of cases.
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