Shadow biquandles and local biquandles

Abstract

Given a shadow biquandle (B,X) composed of a biquandle B and a strongly connected B-set X, we have a local biquandle structure on X. The (co)homology groups of such shadow biquandles are isomorphic to those of the corresponding local biquandles. Moreover, cocycle invariants, of oriented links and oriented surface-links, using such shadow biquandles coincide with those using the corresponding local biquandles. These results imply that for some cases, the Niebrzydowski's theory in [12–14] for knot-theoretic ternary quasigroups is the same as shadow biquandle theory. We also show that some local biquandle 2- or 3-cocycles and some 1- or 2-cocycles of the Niebrzydowski's (co)homology theory can be induced from Mochizuki's (bi)quandle cocycles.