Bilinear Enhancements of Quandle Invariants

Will Gilroy; Sam Nelson

Home/Archive/arXiv

arXiv · 2206.13377

Bilinear Enhancements of Quandle Invariants

Will Gilroy·Sam Nelson

View original source ↗Read source document ↗Export metadata ↓

Abstract

We enhance the quandle counting invariants of oriented classical and virtual knots and links using a construction similar to quandle modules but inspired by symplectic quandle operations rather than Alexander quandle operations. Given a finite quandle $X$ and a vector space $V$ over a field, sets of bilinear forms on $V$ indexed by pairs of elements of $X$ satisfying certain conditions yield new enhanced multiset- and polynomial-valued invariants of oriented classical and virtual knots and links. We provide examples to illustrate the computation of the invariants and to show that the enhancement is proper.

Explore related subjects

math.GT math.QA

Keep this discovery

BibTeX RIS

Will Gilroy, Sam Nelson. 2022-06-27. Bilinear Enhancements of Quandle Invariants. https://arxiv.org/abs/2206.13377

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

KEEP EXPLORING

Related papers

arXiv2026

Bar cohomology of links: beyond Milnor invariants

We develop bar cohomology of link complements as an invariant of links in homology spheres. In this setting, bar cohomology is a Hopf algebra which is calculable using surfaces and their intersection curves in a link complement. In this first in a sequence of works, we introduce the invariant and show that it defines a canonical subspace of the tensor Hopf algebra, which already encodes information about Milnor's link invariants and provides geometrically significant information beyond them.

math.GT↗

arXiv2026

A counterexample to the wrapping number conjecture

We exhibit an annular knot with wrapping number four whose Kauffman bracket has annular degree at most two. This disproves the wrapping number conjecture.

math.GT↗

arXiv2026

Homological lifts of Arnold invariants $J^-$ and $J^+$

Viro's Euler-integral polynomial $P_C(q)$ and the Lanzat--Polyak quantized-curvature polynomial $I_q(C)$ refine Arnold's invariants $J^-$ and $J^+$ for generic immersed one-component plane curves. We construct homological lifts of both. The bigraded region homology retains the singular homology of every connected Alexander-index region; its graded Euler characteristic is $P_C(q)$. The triply graded smoothing-circle homology is generated by the oriented circles of the orientation-preserving smoothing and decategorifies to the smoothing term in $I_q(C)$. Keeping the actual region summands and the boundary regions of every smoothing circle gives a homological refinement of the oriented smoothing configuration, or Seifert state. An infinite family proves strictness: both polynomial data and the ordinary homological lifts agree, while the component-graded region homology and the branch-decomposed circle homology distinguish every pair. Further constructions recover the full $I_q(C)$ by a vertex complex, realize the local change of its curvature integral by edge homology, and give a canonical two-state homology for unoriented curves. Viro described his Euler-integral formula as an analogue of face state-sum formulas for quantum knot polynomials. Through the categorifications developed here, we obtain one concrete homological face-state-sum model realizing that analogy.

math.GT↗