Yang-Baxter operators arising from algebra structures and the Alexander polynomial of knots

Polynomial knot and link invariants from the virtual biquandle

The Alexander biquandle of a virtual knot or link is a module over a 2-variable Laurent polynomial ring which is an invariant of virtual knots and links. The elementary ideals of this module are then invariants of virtual isotopy which determine both the generalized Alexander polynomial (also known as the Sawollek polynomial) for virtual knots and the classical Alexander polynomial for classical knots. For a fixed monomial ordering $<$, the Gröbner bases for these ideals are computable, comparable invariants which fully determine the elementary ideals and which generalize and unify the classic…

Virtual Yang-Baxter cocycle invariants

We extend the Yang-Baxter cocycle invariants for virtual knots by augmenting Yang-Baxter 2-cocycles with cocycles from a cohomology theory associated to a virtual biquandle structure. These invariants coincide with the classical Yang-Baxter cocycle invariants for classical knots but provide extra information about virtual knots and links. In particular, they provide a method for detecting non-classicality of virtual knots and links.

扭曲Yang-Baxter集、上同调理论及其在纽结中的应用

Twisted Yang-Baxter sets, cohomology theory, and application to knots

该研究引入扭曲Yang-Baxter集并构建拓展的上同调理论，结合扭曲双广群上循环与Alexander编号构造纽结及曲面的状态和不变量，成功区分2-扭转纺三叶结与其反向构型，验证了方法有效性。

Generalizations of Quandle Cocycle Invariants and Alexander Modules from Quandle Modules

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.

纽结群在$\textrm{AGL}_{1}(\mathbb{C})$中的表示与Alexander不变量

Representations of knot groups in $\textrm{AGL}_{1}(\mathbb{C})$ and Alexander invariants

本文通过纽结群在复仿射群AGL1(C)中的表示簇重新诠释Alexander不变量，证明表示簇坐标环与Alexander模对称代数同构，并构造拓扑量子场论以提供计算方法。

Finite Type Invariants of w-Knotted Objects: From Alexander to Kashiwara and Vergne

This preprint was split in two and became the first two parts of a four-part series (arXiv:1405.1956, arXiv:1405:1955, and two in preparation). The remaining relevance of this preprint is due to the series of videotaped lectures (wClips) that are linked within.

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.

Finite Type Invariants of w-Knotted Objects I: w-Knots and the Alexander Polynomial

This is the first in a series of papers studying w-knotted objects (w-knots, w-braids, w-tangles, etc.), which make a class of knotted objects which is {w}ider but {w}eaker than their usual counterparts. The group of w-braids was studied (as "{w}elded braids") by Fenn-Rimanyi-Rourke and was shown to be isomorphic to the McCool group of "basis-conjugating" automorphisms of a free group Fn. Brendle-Hatcher, tracing back to Goldsmith, have shown this group to be a group of movies of flying rings in R3. Satoh studied several classes of w-knotted objects (as "{w}eakly-virtual") and has shown them t…

Three-dimensional BF Theories and the Alexander-Conway Invariant of Knots

We study 3-dimensional BF theories and define observables related to knots and links. The quantum expectation values of these observables give the coefficients of the Alexander-Conway polynomial.

基于杨-巴克斯特上同调的纽结量子上循环不变量

Quantum Cocycle Invariants of Knots from Yang-Baxter Cohomology

本文基于杨-巴克斯特上同调的2-上循环，从形变理论视角引入纽结量子上循环不变量，证明其可解释拟群上循环不变量且在部分例子中更强，还推导了琼斯、亚历山大多项式并给出相关实例。

从量子sl_3出发的非阿贝尔化亚历山大多项式

A Non-Abelian Generalization of the Alexander Polynomial from Quantum $\mathfrak{sl}_3$

本文提出一种非阿贝尔化的亚历山大多项式，通过量子sl_3的表示在四次单位根下的构造，建立了一个针对半单李代数的链接不变量，并展示了其与亚历山大多项式的关系。

The LMO-invariant of 3-manifolds of rank one and the Alexander polynomial

We prove that the LMO-invariant of a 3-manifold of rank one is determined by the Alexander polynomial of the manifold, and conversely, that the Alexander polynomial is determined by the LMO-invariant. Furthermore, we show that the Alexander polynomial of a null-homologous knot in a rational homology 3-sphere can be obtained by composing the weight system of the Alexander polynomial with the Aarhus invariant of knots.

The Alexander- and Jones-invariants and the Burau module

From the braid-valued Burau module over the braid group we construct the Yang-Baxter matrices yielding the Alexander- and the Jones knot invariants. This generalises an observation of V. F. R. Jones.

Random Walk on Knot Diagrams, Colored Jones Polynomial and Ihara-Selberg Zeta Function

A model of random walk on knot diagrams is used to study the Alexander polynomial and the colored Jones polynomial of knots. In this context, the inverse of the Alexander polynomial of a knot plays the role of an Ihara-Selberg zeta function of a directed weighted graph, counting with weights cycles of random walk on a 1-string link whose closure is the knot in question. The colored Jones polynomial then counts with weights families of ``self-avoiding'' cycles of random walk on the cabling of the 1-string link. As a consequence of such interpretations of the Alexander and colored Jones polynomi…

Alexander-Beck modules detect the unknot

We introduce the Alexander-Beck module of a knot as a canonical refinement of the classical Alexander module, and we prove that this new invariant is an unknot-detector.

Universal weight systems and the Melvin-Morton expansion of the colored Jones knot invariant

We study the asymptotic expansion of the colored Jones polynomial (the Melvin-Morton expansion) using a recursion formula for the deframed universal weight system for the $sl(2)$ Lie algebra. Combined with the formula for the universal weight system for the Lie superalgebra $gl(1|1)$ (which corresponds to the Alexander-Conway knot polynomial) this formula gives a very short proof of the Melvin-Morton conjecture relating the colored Jones invariant and the Alexander-Conway polynomial of knots.

New Link Invariants and Yang-Baxter Equation

We have new solutions to the Yang-Baxter equation, from which we have constructed new link invariants containing more than two arbitrary parameters. This may be regarded as a generalization of the Jones' polynomial. We have also found another simpler invariant which discriminates only the linking structure of knots with each other, but not details of individual knot.