Quantum enhancements and biquandle brackets

Abstract

We introduce a new class of quantum enhancements we call biquandle brackets, which are customized skein invariants for biquandle colored links. Quantum enhancements of biquandle counting invariants form a class of knot and link invariants that includes biquandle cocycle invariants and skein invariants such as the HOMFLY-PT polynomial as special cases, providing an explicit unification of these apparently unrelated types of invariants. We provide examples demonstrating that the new invariants are not determined by the biquandle counting invariant, the knot quandle, the knot group or the traditional skein invariants.

Picture-valued parity-biquandle bracket

“…It is worth saying that classical knot theory embeds in virtual knot theory [12,17] and this fact is not trivial. [1,2,7,8,20,21] is a set X with two binary operations •, * : X × X → X satisfying the following axioms:…”

“…As a result, we get the following definitions (cf. [20]). Let L be an oriented (virtual) link diagram with n crossings and let…”

“…The beautiful idea due to Nelson, Orrison, Rivera [20,21] allows one to use colours or (bi)quandles in order to enhance various invariants of knots. To this end, one takes a colouring of a knot diagram and a quantum invariant which satisfies a certain skein-relation (say, Kauffman bracket) and tries to take different coefficients for the states: we take into account the colours of the (short) arcs of the knot 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].…”

“…Hence we have the following theorem (see [6]): yields [(5)(4)6 2 + (1)(4)6 + (5)(5)6 + (1)(5)6 2 ]2 −2 = (6 + 3 + 3 + 5)2 = 6; yields [(4)(5)6 2 + (5)(5)6 + (4)(1)6 + (5)(1)6 2 ]2 −2 = (6 + 3 + 3 + 5)2 = 6, and yields [(1)(1)6 2 + (3)(1)6 + (1)(3)6 + (3)(3)6 2 ]2 −2 = (1 + 4 + 4 + 2)2 = 1. Then the multiset form of the invariant is Φ β,M X (L) = {1, 1, 6, 6}, or in polynomial form we have Φ β X (L) = 2u + 2u 6 .…”