Oshiro Kanako

This is the second paper which discusses minimum numbers of “region” colors for knots, while minimum numbers of arc colors are well-studied, where the first one is our previous paper [6]. In this paper, we give a method to evaluate minimum numbers of Dehn colors for knots by using symmetric local biquandle cocycle invariants. We give answers to some questions arising as a consequence of our previous paper [6]. In particular, we show that there exist knots which are distinguished by minimum numbers of Dehn colors.

The purpose of this paper is to introduce an enriched presentation, called a shade quandle presentation, containing information that can be used to normalize Alexander type invariants. We also introduce transformations on shade quandle presentations and show that two shade quandle presentations of an oriented link are related by the transformations. We see that the transformations are finer than the strong Tietze transformations to normalize Alexander type invariants.

In this paper, we introduce the (co)homology group of a multiple conjugation biquandle. It is the (co)homology group of the prismatic chain complex, which is related to the homology of foams introduced by J. S. Carter, modulo a certain subchain complex. We construct invariants for [Formula: see text]-oriented handlebody-links using [Formula: see text]-cocycles. When a multiple conjugation biquandle [Formula: see text] is obtained from a biquandle [Formula: see text] using [Formula: see text]-parallel operations, we provide a [Formula: see text]-cocycle (or [Formula: see text]-cocycle) of the multiple conjugation biquandle [Formula: see text] from a [Formula: see text]-cocycle (or [Formula: see text]-cocycle) of the biquandle [Formula: see text] equipped with an [Formula: see text]-set [Formula: see text].