Search arXiv⌕ Search

arXiv · 0708.1951

On bilinear biquandles

Abstract

We define a type of biquandle which is a generalization of symplectic quandles. We use the extra structure of these bilinear biquandles to define new knot and link invariants and give some examples.

Explore related subjects

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Sam Nelson, Jacquelyn L. Rische. 2007-12-21. On bilinear biquandles. https://arxiv.org/abs/0708.1951

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

Classification of finite-dimensional Nichols algebras of rank two in twisted Yetter--Drinfeld categories

Let \(G\) be a finite non-abelian group, let \(Φ\in Z^3(G,\mathbb C^\times)\) be normalized, and let \(V,W\) be finite-dimensional simple objects of \({}_G^G\mathcal{YD}^Φ\). We classify, up to interchange, the braided-indecomposable pairs \((V,W)\) whose supports generate \(G\) and for which \(\mathcal B(V\oplus W)\) is finite-dimensional. The classification consists of eight cases with five possible support quandles. In every case the Cartan graph is standard of type \(A_2\), \(B_2\), or \(G_2\), and the dimension is determined explicitly. A new phenomenon occurs {in the \(Γ_2\) case}: the twisted setting admits a family of type \(G_2\) absent from the ordinary \(Γ_2\) classification and we construct an explicit example over a non-abelian group of order \(16\).

math.QA↗

$K_{\mathfrak{q}}$-algebras

We introduce the notion of $K_{\mathfrak{q}}$-algebra, a conjectural axiomatization of the fusion algebra of rotation-equivariant BPS line defects in a 4d ${\cal N}=2$ Supersymmetric Quantum Field Theory. We also introduce the notion of RG flow of $K_{\mathfrak{q}}$-algebras, a conjectural axiomatization of the action of Seiberg-Witten type RG flows on line defects. These definitions enrich the theory of cluster algebras, $K$-theoretic Coulomb branch algebras, skein algebras and quantum groups.

math.QA↗

Outer action of representation category of discrete quantum groups: a case study

Given any tracial von Neumann algebra $\mathcal{A}$, the first author has defined a unitary tensor functor $Φ:{\rm Rep}(\mathcal{Q}_{\rm aut}(\mathcal{A}))\to {\rm Bimod}(\mathcal{A})$ in his paper [Gos24]. In this paper, we consider $\mathcal{A}$ to be the underlying von Neumann algebra of $C^*(S_3)$ and show that given any DQG $\mathcal{Q}$, which contains $C^*(S_3)$ as a quantum subgroup and has an outer action $\tildeΦ$ (in the sense of Definition 3.4}) on ${\rm Bimod}(\mathcal{A})$, which agrees with $Φ$ when restricted on ${\rm Rep}(C^*(S_3))$, is monoidally equivalent to the DQG $C^*(S_3)$. Furthermore, given any DQG $\mathcal{Q}$ with an outer action $Γ:{\rm Rep}(\mathcal{Q})\to {\rm Bimod}(\mathcal{A})$, such that $ {\rm Out}(\mathcal{A})\subseteq {\rm Im}(Γ)$, is either monoidally equivalent to Out$(\mathcal{A})$ or monoidally equivalent to $C^*(S_3)$ itself.

math.QA↗