Polynomial invariants of tribrackets in knot theory (Q6046554)

One of the main problems in knot theory is the knot recognition problem. The essence of this problem is to construct an effective algorithm that determines by two given knots whether they are equivalent or not. One possible approach to this problem is to construct invariants, i.e. such functions that have the same value on equivalent knots. If we manage to find an invariant \(f\) such that \(f(K_1)\neq f(K_2)\), then we can conclude that \(K_1\) and \(K_2\) are non-equivalent knots. One of the strongest invariants in knot theory is the quandle invariant \(K\to Q(K)\), which for a given knot \(K\) gives an algebraic system \(Q(K)\) called quandle. Quandles were introduced in 1982 by \textit{D. Joyce} [J. Pure Appl. Algebra 23, 37--65 (1982; Zbl 0474.57003)] and \textit{S. V. Matveev} [Math. USSR, Sb. 47, 73--83 (1984; Zbl 0523.57006); translation from Mat. Sb., Nov. Ser. 119(161), No. 1, 78--88 (1982)]. Over the years, quandles have been investigated by various authors for constructing newer invariants for knots and links. In [\textit{S. Nelson}, J. Algebra Appl. 7, No. 2, 263--273 (2008; Zbl 1152.57015)], a two-variable polynomial invariant of finite quandles known as the quandle polynomial was introduced. Sets with ternary operations known as knot-theoretic ternary quasigroups were introduced in [\textit{M. Niebrzydowski} et al., Fundam. Math. 247, No. 3, 299--320 (2019; Zbl 1457.20055)] and used to define knot and link invariants. By the analogy to the quandle polynomial from [\textit{S. Nelson}, loc. cit.] the authors of the paper under review introduce a notion of tribracket polynomial. As an application, the authors enhance the tribracket counting invariant of knots and links using subtribracket polynomials and provide examples to demonstrate that this enhancement is proper.