Virtual tribrackets

Abstract

We introduce virtual tribrackets, an algebraic structure for coloring regions in the planar complement of an oriented virtual knot or link diagram. We use these structures to define counting invariants of virtual knots and links and provide examples of the computation of the invariant; in particular we show that the invariant can distinguish certain virtual knots.

Knot-theoretic ternary groups

“…In [11] a strategy was used of utilizing two ternary operations, satisfying mixed axioms, for oriented virtual links. One operation was for the classical crossings, and the other for the virtual ones.…”

Tribracket Modules

“…A related structure called biquasiles was introduced in [13] by two of the authors with applications to surface-links in [11] by one of the authors. Recently ternary quasigroup operations known as Niebrzydowski tribrackets have been studied with additional generalizations to the cases of virtual knots in [15] and trivalent spatial graphs in [8].…”

Multi-tribrackets

“…In [11], tribracket colorings were extended to virtual knots and links and in [4], tribrackets were enhanced with partial products to define Niebrzydowski algebras, algebraic structures for coloring Y -oriented spatial graphs and handlebody-links.…”

“…In this paper we define multi-tribrackets, a generalization of the tribracket structure which includes virtual tribrackets from [11] as a special case and which defines stronger counting invariants for virtual knots and links with multiple components. The paper is organized as follows.…”

“…We provide examples to show that the resulting counting invariants can distinguish links which are not distinguished by the counting invariants associated to the standard tribracket coloring. We reinterpret the results of [11] in terms of multi-tribrackets and consider future directions for multi-tribracket theory.…”