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Vaughan Jones was my mentor and friend. I provide here a few personal recollections including the “knotted antennas” project which brought me to Berkeley. [ABSTRACT FROM AUTHOR]

W. Rump showed that there exists a one-to-one correspondence between involutive right non-degenerate solutions of the Yang–Baxter equation and cycle sets. J. S. Carter, M. Elhamdadi, and M. Saito, meanwhile, introduced a homology theory of set-theoretic solutions of the Yang–Baxter equation in order to define cocycle invariants of classical knots. In this paper, we introduce the normalized homology theory of an involutive right non-degenerate solution of the Yang–Baxter equation and compute the normalized set-theoretic Yang–Baxter homology of cyclic racks. Moreover, we explicitly calculate some two-cocycles, which can be used to classify certain families of torus links. [ABSTRACT FROM AUTHOR]

Based on the presentation of the Kauffman bracket skein algebra of the thickened torus given by the third author in previous work [4], Frohman and Gelca established a complete description of the multiplicative operation leading to a famous product-to-sum formula. In this paper, we study the multiplicative structure of the Kauffman bracket skein algebra of the thickened four-holed sphere. We present an algorithm to compute the product of any two elements of the algebra, and give an explicit formula for some families of curves. We surmise that the algorithm has quasi-polynomial growth with respect to the number of crossings of a pair of curves. Further, we conjecture the existence of a positive basis for the algebra. [ABSTRACT FROM AUTHOR]

In this paper, we adjust the Yang–Baxter operators constructed by Jones for the HOMFLYPT polynomial. Then we compute the second homology for this family of Yang–Baxter operators. It has the potential to yield 2-cocycle invariants for links. [ABSTRACT FROM AUTHOR]

Conference like this in Dallas is a perfect occasion to give some reminiscences of my 40 years of work as a mathematician, and to muse a little about my mathematical past. It is a rare occasion to be a little sentimental. In this essay, I summarize and expand my birthday talk given at University of Texas at Dallas conference. There, I mostly covered the early years of my mathematical career ending with a detailed description about the smallest volume hyperbolic 3-manifold and the HOMFLYPT polynomial. Thus I seldom venture to the time after 1984 (for example to mention skein modules constructed in April 1987) and only occasionally after 1995 when I settled to work at George Washington University and have had several successful PhD students. [ABSTRACT FROM AUTHOR]

We show that the Kauffman bracket skein algebra of any oriented surface F (possibly with marked points in its boundary) has no zero divisors and that its center is generated by knots parallel to the unmarked components of the boundary of F. Furthermore, we show that skein algebras are Noetherian and Ore. Our proofs rely on certain filtrations of skein algebras induced by pants decompositions of surfaces. We prove some basic algebraic properties of the associated graded algebras along the way. [ABSTRACT FROM AUTHOR]

In 2004, Carter, Elhamdadi and Saito defined a homology theory for set-theoretic Yang–Baxter operators (we will call it the “algebraic” version in this paper). In 2012, Przytycki defined another homology theory for pre-Yang–Baxter operators which has a nice graphic visualization (we will call it the “graphic” version in this paper). We show that they are equivalent. The “graphic” homology is also defined for pre-Yang–Baxter operators, and we give some examples of its one-term and two-term homologies. In the two-term case, we have found torsion in homology of Yang–Baxter operator that yields the Jones polynomial. [ABSTRACT FROM AUTHOR]

We classify rooted trees which have strictly unimodal plucking polynomials ( q -polynomials). We also give criteria for a trapezoidal shape of a plucking polynomial. We generalize results of Pak and Panova on strict unimodality of q -binomial coefficients. [ABSTRACT FROM AUTHOR]

We give necessary and sufficient conditions for a given polynomial to be a plucking polynomial of a rooted tree. We discuss the fact that different rooted trees can have the same polynomial. [ABSTRACT FROM AUTHOR]

It is a classical result in reduced homology of finite groups that the order of a group annihilates its homology. Similarly, we have proved that the torsion subgroup of rack and quandle homology of a finite quasigroup quandle is annihilated by its order. However, it does not hold for connected quandles in general. In this paper, we define an -almost quasigroup (-AQ) quandle which is a generalization of a quasigroup quandle and study annihilation of torsion in its rack and quandle homology groups. [ABSTRACT FROM AUTHOR]

We show that every alternating link of two components and crossings can be reduced by -moves to the trivial link or the Hopf link. It answers the question asked in one of the last papers by Slavik Jablan. [ABSTRACT FROM AUTHOR]

We prove that if G is an abelian group of odd order then there is an isomorphism from the second quandle homology $H_2^Q(T(G))$ to G ∧ G, where ∧ is the exterior product. In particular, for $G=\mathbb{Z}_k^n$, k odd, we have $H_2^Q(T(\mathbb{Z}_k^n)) = \mathbb{Z}_k^{n(n-1)/2}$. Nontrivial $H_2^Q(T(G))$ allows us to use 2-cocycles to construct new quandles from T(G), and to construct link invariants. Computation of $H_2^Q(T(G))$ is also the first, fundamental step in the direction of computing homology of Takasaki quandles in general. [ABSTRACT FROM AUTHOR]

We consider the problem of classification of links up to (2, 2)-moves. Our motivation comes from the theory of skein modules, more specifically from the skein module of S3 based on the deformation of (2, 2)-move. As it was proved in D-P-2, not every link can be reduced to a trivial link by (2, 2)-moves, for instance, the closure of (σ1σ2)6. In this paper, we classify 3-braids up to (2, 2)-moves and, we show how the Harikae–Nakanishi–Uchida conjecture can be modified to hold for closed 3-braids. As an important step in the classification we prove the conjecture for 2-algebraic links and classify (2, 2)-equivalence classes for links up to nine crossings. We also analyze an action of (2, 2)-move on Kei (involutive quandle) associated to a link diagram. We define Burnside Kei, Q(m, n), and ask for which values of m and n, is Q(m, n) finite. This question is motivated by classical Burnside problem. [ABSTRACT FROM AUTHOR]

We analyze properties of links which have diagrams with a small number of negative crossings. We show that if a nontrivial link has a diagram with all crossings positive except possibly one, then the signature of the link is negative. If a link diagram has two negative crossings, we show that the signature of the link is nonpositive with the exception of the left-handed Hopf link (possibly, with extra trivial components). We also characterize those links which have signature zero and diagrams with two negative crossings. In particular, we show that if a nontrivial knot has a diagram with two negative crossings then the signature of the knot is negative, unless the knot is a twist knot with negative clasp. We completely determine all trivial link diagrams with two or fewer negative crossings. For a knot diagram with three negative crossings, the signature of the knot is nonpositive except for the left-handed trefoil knot. These results generalize those of Rudolph, Cochran, Gompf, Traczyk and Przytycki, solve [27, Conjecture 5], and give a partial answer to [3, Problem 2.8] about knots dominating the trefoil knot or the trivial knot. We also describe all unknotting number one positive knots. [ABSTRACT FROM AUTHOR]