Picture-valued parity-biquandle bracket

Abstract

In [3] we constructed the parity-biquandle bracket valued in pictures (linear combinations of 4valent graphs). We gave no example of classical links such that the parity-biquandle bracket of which is not trivial.In the present paper we slightly change the notation of the parity-biquandle bracket and give examples of knots and links having a non-trivial parity-biquandle bracket. As a result we get the minimality theorem. This is the first evidence that graphs (link shadows) appear as invariants of link diagrams instead of just polynomials groups and other tractable objects.1) it is defined by using states in a way similar to the Kauffman bracket, 2) it is valued not in numbers or (Laurent) polynomials but in diagrams meaning that we do not completely resolve a knot diagram leaving some crossings intact.It is important to note that for some (completely odd) diagrams, no crossings are smoothed at all. It is the first appearance of diagram-valued invariants in knot theory. For virtual knots, it allows one to make very strong conclusions about the shape of any diagram by looking just at one diagram. Unfortunately, the value of the bracket at classical knots is trivial.

The topological biquandle of a link

“…virtual links), however, we sometimes need to combine two or more different link invariants to yield a stronger invariant. Some examples of this are the quantum enhancements using biquandles, see [12,13,14,9]. In the study of virtual links, Manturov introduced the concept of parity [10], that induces a function on the set of crossings of any virtual link diagram.…”

A Survey of Quantum Enhancements

“…A biquandle bracket is a skein invariant for biquandle-colored knots and links. The definition was introduced in [6] (and independently, a special case was introduced in [1]) and has only started to be explored in other recent work such as [8,7,4].…”

“…In [4], another skein relation is used in the biquandle bracket setting, superficially similar to the biquandle virtual brackets described above but with the virtual smoothing replaced with a 4-valent vertex.…”

“…• The first approach for generalization, examples of which have been considered in [7] and [4], is to apply the biquandle bracket idea to different skein relations. One may find that skein relations which do not yield anything new in the uncolored case can provide new invariants in various biquandle-colored cases.…”

“…A special case of biquandle brackets was described independently in [1]. In [4] a type of biquandle bracket whose skein relation includes a vertex is considered. In [8] biquandle brackets are defined using trace diagrams in order to allow for recursive expansion as opposed to the state-sum definition in [6].…”