Quantum Enhancements via Tribracket Brackets
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Quantum Enhancements via Tribracket Brackets Laira Aggarwal, Sam Nelson and Patricia Rivera Abstract. We enhance the tribracket counting invariant with tribracket brackets, skein invariants of tribracket-colored oriented knots and links analogously to biquandle brackets. This infinite family of invariants includes the classical quantum invariants and tribracket cocycle invariants as special cases, as well as new invariants. Wex provide explicit examples as well as questions for future work. Mathematics Subject Classification. 57M27, 57M25.
1. Introduction In recent decades, much work has involved colorings of knot and link diagrams by algebraic structures such as quandles and biquandles. These colorings have an interpretation as elements of a homset collecting homomorphisms from an object associated with the knot or link such as the knot group and the knot quandle to the finite coloring object. This interpretation gives rise to the notion of enhancements, stronger invariants obtained from the set of colorings by collecting values of an invariant of colored knots, starting with quandle cocycle invariants of [2] and explored in many subsequent works. See [3,5] for an overview. In [11], the second-listed author and a coauthor proposed the notion of quantum enhancements, quantum invariants of biquandle-colored knots and links. In [7], the second-listed author and coauthors introduced a class of quantum enhancements known as biquandle brackets, skein invariants of biquandle-colored knots and links. This infinite class of invariants includes the classical quantum invariants (Alexander–Conway, Jones, HOMFLYPT and Kauffman polynomials) and the quandle cocycle invariants as special cases, explicitly uniting these a priori unrelated invariants; moreover, examples of biquandle brackets which are apparently neither of these types were identified in [7]. In [9], biquandle brackets were extended to define biquandle virtual Partially supported by Simons Foundation Collaboration Grant 316709.
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brackets using a skein relation including a virtual crossing, in [4], a type of biquandle bracket with skein relation involving graphs was introduced, and in [10] a method of computing biquandle bracket invariants recursively with trace diagrams as opposed to via the original state-sum definition was introduced. See [6] for a survey of biquandle bracket invariants. In [12], colorings of knot diagrams by knot-theoretic ternary quasigroups, or as we call them Niebrzydowski tribrackets, were introduced. In this paper, we extend the biquandle bracket idea to the case of tribrackets, defining quantum enhancements via tribracket brackets. The paper is organized as follows. In Sect. 2, we review the basics of tribracket theory. In Sect. 3, we introduce tribracket brackets and provide some examples, including an illustration of the computation of the new invariant, examples showing that the new invariant is proper enhancement, and the values of the tribracket bracket invariant for two bra
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