TL;DR
This paper introduces finite type enhancements for biquandle invariants using truncated biquandle-labeled Polyak algebras, revealing new nontrivial invariants related to biquandle cocycle invariants.
Contribution
It extends biquandle counting invariants with finite type enhancements based on Polyak algebras, generalizing previous invariants and establishing new connections.
Findings
Finite type enhancements reduce to known invariants for trivial biquandles.
Biquandle labeled finite type invariants of degree 1 are nontrivial.
New relations to biquandle cocycle invariants are established.
Abstract
We enhance the biquandle counting invariant using elements of truncated biquandle-labeled Polyak algebras. These finite type enhancements reduce to the finite type enhancements defined by Goussarov, Polyak and Viro for the trivial biquandle of one element and determine (but are not determined by) the biquandle counting invariant for general biquandles. Unlike the unlabeled case, biquandle labeled finite type invariants of degree 1 are nontrivial and are related to biquandle cocycle invariants.
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Taxonomy
TopicsGeometric and Algebraic Topology · Advanced Combinatorial Mathematics · semigroups and automata theory