Matrices and Finite Biquandles

双结点模箭图表示

Biquandle Module Quiver Representations

本文提出了一种与有限双结点等结构相关的经典、虚拟和曲面-纽结与链环的无限族箭图表示值不变量，并以此定义了新的双变量多项式不变量无限族。

虚拟双quandle上同调箭图表示

Virtual Biquandle Cocycle Quiver Representations

本文针对定向虚拟纽结与链环，基于有限虚拟双quandle等结构引入箭图表示值不变量，并通过去范畴化得到新的无限族多项式不变量，为虚拟纽结理论提供了新的不变量构造方法。

基于双quandle的浸入曲面链不变量、吉川定向第五变换与带边2-结

On biquandle-based invariant of immersed surface-links, Yoshikawa oriented fifth move, and ribbon 2-knots

该研究解决了吉川定向第五变换不可被其他变换替代的开放问题，扩展了浸入曲面链的代数工具包并开发了双quandle着色理论，还证明了无限多对群同构但结quandle不同的带边2-结。

广义双架与双架多项式不变量

Generalised Biracks and the Birack Polynomial Invariant

本文将有限双量子着色不变量推广为基于有限双架标记纽结图的多项式不变量，并借此重新梳理相关广义纽结理论与广义双架理论。

Link invariants from finite biracks

A birack is an algebraic structure with axioms encoding the blackboard-framed Reidemeister moves, incorporating quandles, racks, strong biquandles and semiquandles as special cases. In this paper we extend the counting invariant for finite racks to the case of finite biracks. We introduce a family of biracks generalizing Alexander quandles, $(t,s)$-racks, Alexander biquandles and Silver-Williams switches, known as $(τ,σ,ρ)$-biracks. We consider enhancements of the counting invariant using writhe vectors, image subbiracks, and birack polynomials.

Skew Brace Enhancements

We use the structure of skew braces to enhance the biquandle counting invariant for virtual knots and links for finite biquandles defined from skew braces. We introduce two new invariants: a single-variable polynomial using skew brace ideals and a two-variable polynomial using the skew brace group structures. We provide examples to show that the new invariants are not determined by the counting invariant and hence are proper enhancements.

弦指标：定义、应用与推广

The chord index, its definitions, applications and generalizations

本文研究虚拟结的弦指标，将其推广为有限型不变量，并引入指标Jones多项式和指标Quandle，讨论其在扭结理论中的应用。

Generalized quandle polynomials

We define a family of generalizations of the two-variable quandle polynomial. These polynomial invariants generalize in a natural way to eight-variable polynomial invariants of finite biquandles. We use these polynomials to define a family of link invariants which further generalize the quandle counting invariant.

Bikei, Involutory Biracks and unoriented link invariants

We identify a subcategory of biracks which define counting invariants of unoriented links, which we call involutory biracks. In particular, involutory biracks of birack rank N=1 are biquandles, which we call bikei. We define counting invariants of unoriented classical and virtual links using finite involutory biracks, and we give an example of a non-involutory birack whose counting invariant detects the non-invertibility of a virtual knot.

Quandle上同调箭图表示

Quandle Cohomology Quiver Representations

本文定义了一族基于有限quandle、阿贝尔群及2-上闭链等参数的箭图表示值纽结与链环不变量，由此导出四个新多项式不变量，并推广至双quandle情形。

Matrices and Finite Quandles

Finite quandles with n elements can be represented as n-by-n matrices. We show how to use these matrices to distinguish all isomorphism classes of finite quandles for a given cardinality n, as well as how to compute the automorphism group of each finite quandle. As an application, we classify finite quandles with up to 5 elements and compute the automorphism group for each quandle.

Finite type invariants of knots via their Seifert matrices

We define a filtration on the vector space spanned by Seifert matrices of knots related to Vassiliev's filtration on the space of knots. Further we show that the invariants of knots derived from the filtration can be expressed by coefficients of the Alexander polynomial.

Matrices and finite Alexander quandles

We describe an algorithm for determining whether a finite quandle is isomorphic to an Alexander quandle by finding all possible Alexander presentations of the quandle. We give an implementation of this algorithm in Maple.