Ternary quasigroups in knot theory

Maciej Niebrzydowski

Published 2017-08-17Version 1

We show that some ternary quasigroups appear naturally as invariants of classical, virtual, and flat virtual links. We also note how to obtain from them invariants of Yoshikawa moves. To deal with virtual crossings, we introduce ternary edge bicolorings. In our previous paper, we defined homology theory for algebras satisfying two axioms derived from the third Reidemeister move. In this paper, we show a degenerate subcomplex suitable for ternary quasigroups satisfying these axioms, and corresponding to the first Reidemeister move. For such ternary quasigroups with an additional condition that the primary operation equals to the second division operation, we also define another subcomplex, corresponding to the flat second Reidemeister move. Based on the normalized homology, we define cocycle invariants.