Partially multiplicative biquandles and handlebody-knots

Abstract

We introduce several algebraic structures related to handlebody-knots, including G-families of biquandles, partially multiplicative biquandles and group decomposable biquandles. These structures can be used to color the semiarcs in Y -oriented spatial trivalent graph diagrams representing S 1 -oriented handlebody-knots to obtain computable invariants for handlebody-knots and handlebody-links. In the case of G-families of biquandles, we enhance the counting invariant using the group G to obtain a polynomial invariant of handlebody-knots.

Biquandle (co)homology and handlebody-links

“…Definition 2.4 ( [11]). Let G be a group with identity element e. A G-family of biquandles is a non-empty set X with two families of binary operations * g , * g : X × X → X (g ∈ G) satisfying the following axioms.…”

“…If X is finite, then (X, ( * [n] ) n∈Ztype X , ( * [n] ) n∈Ztype X ) is a Z type X -family of biquandles [11].…”

“…The idea of quandle colorings of spatial graphs using flows was generalized to colorings using G-family of quandles in [9] and colorings using multiple conjugation quandles in [6]. The notion of a G-family of quandles is generalized to a G-family of biquandles and the notion of a partially multiplicative biquandle was introduced in [11]. In our previous paper [10], we introduced a multiple conjugation biquandle as a generalization of a multiple conjugation quandle, and proved that it is the universal algebra for defining a semi-arc coloring invariant for handlebody-links.…”

The Gordian distance of handlebody-knots and Alexander biquandle colorings

“…Definition 3.5 ( [10,13]). Let G be a group with the identity element e. A G-family of biquandles is a non-empty set X with two families of binary operations * g , * g : X × X → X (g ∈ G) satisfying the following axioms.…”

“…Ishii, Iwakiri, Jang and Oshiro [9] introduced a G-family of quandles, which is an extension of the above structures. Recently, Ishii and Nelson [13] introduced a G-family of biquandles, which is a biquandle version of a G-family of quandles.…”