Link invariants from finite racks
Sam Nelson
TL;DR
This paper introduces new ambient isotopy invariants for knots and links derived from finite racks, extending quandle invariants and enhancing them with rack cohomology 2-cocycles.
Contribution
It generalizes quandle counting invariants to finite racks and introduces a new degeneracy condition for rack cohomology enhancements.
Findings
Invariants reduce to quandle counting invariants for quandles.
Enhanced invariants incorporate rack 2-cocycles with a new degeneracy condition.
Provides a framework for new knot and link invariants using rack cohomology.
Abstract
We define ambient isotopy invariants of oriented knots and links using the counting invariants of framed links defined by finite racks. These invariants reduce to the usual quandle counting invariant when the rack in question is a quandle. We are able to further enhance these counting invariants with 2-cocycles from the coloring rack's second rack cohomology satisfying a new degeneracy condition which reduces to the usual case for quandles.
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Taxonomy
TopicsGeometric and Algebraic Topology · Homotopy and Cohomology in Algebraic Topology · Advanced Combinatorial Mathematics