Algebraic versus geometric categorification of the~Alexander polynomial: a~spectral sequence

The Alexander module, Seifert forms, and categorification

We show that bordered Floer homology provides a categorification of a TQFT described by Donaldson. This, in turn, leads to a proof that both the Alexander module of a knot and the Seifert form are completely determined by Heegaard Floer theory.

A quantum categorification of the Alexander polynomial

Using a modified foam evaluation, we give a categorification of the Alexander polynomial of a knot. We also give a purely algebraic version of this knot homology which makes it appear as the infinite page of a spectral sequence starting at the reduced triply graded link homology of Khovanov--Rozansky.

A categorification of the Alexander polynomial in embedded contact homology

Given a transverse knot $K$ in a three dimensional contact manifold $(Y,α)$, in [13] Colin, Ghiggini, Honda and Hutchings define a hat version of embedded contact homology for $K$, that we call $\widehat{ECK}(K,Y,α)$, and conjecture that it is isomorphic to the knot Floer homology $\widehat{HFK}(K,Y)$. We define here a full version $ECK(K,Y,α)$ and generalise the definitions to the case of links. We prove then that, if $Y = S^3$, $ECK$ and $\widehat{ECK}$ categorify the (multivariable) Alexander polynomial of knots and links, obtaining expressions analogue to that for knot and link Floer homol…

Généralisation de l'homologie d'Heegaard-Floer aux entrelacs singuliers & Raffinement de l'homologie de Khovanov aux entrelacs restreints

A categorification of a polynomial link invariant is an homological invariant which contains the polynomial one as its graded Euler characteristic. This field has been initiated by Khovanov categorification of the Jones polynomial. Later, P. Ozsvath and Z. Szabo gave a categorification of Alexander polynomial. Besides their increased abilities for distinguishing knots, this new invariants seem to carry many geometrical informations. On the other hand, Vassiliev works gives another way to study link invariant, by generalizing them to singular links i.e. links with a finite number of rigid trans…

Pretzel 链环 $\oldsymbol{P(1,1,n)}$ 的 Alexander-Conway 与 Bracket 多项式

Alexander-Conway and Bracket Polynomials of Pretzel Links $\boldsymbol{P(1,1,n)}$

本文研究三股 pretzel 链环 P(1,1,n) 的 Alexander-Conway 与 Kauffman bracket 多项式，重新考察这两个经典多项式不变量，以深化对其拓扑和代数性质的理解。

Knots and distributive homology: from arc colorings to Yang-Baxter homology

This paper is a sequel to my essay "Distributivity versus associativity in the homology theory of algebraic structures" Demonstratio Math., 44(4), 2011, 821-867 (arXiv:1109.4850 [math.GT]). We start from naive invariants of arc colorings and survey associative and distributive magmas and their homology with relation to knot theory. We outline potential relations to Khovanov homology and categorification, via Yang-Baxter operators. We use here the fact that Yang-Baxter equation can be thought of as a generalization of self-distributivity. We show how to define and visualize Yang-Baxter homology…