Narrow Results

We consider two approaches to isotopy invariants of oriented links: one from ribbon categories and the other from generalized Yang–Baxter (gYB) operators with appropriate enhancements. The gYB-operators we consider are obtained from so-called gYBE objects following a procedure of Kitaev and Wang. We show that the enhancement of these gYB-operators is canonically related to the twist structure in ribbon categories from which the operators are produced. If a gYB-operator is obtained from a ribbon category, it is reasonable to expect that two approaches would result in the same invariant. We prove that indeed the two link invariants are the same after normalizations. As examples, we study a new family of gYB-operators which is obtained from the ribbon fusion categories SO(N)₂, where N is an odd integer. These operators are given by 8 × 8 matrices with the parameter N and the link invariants are specializations of the two-variable Kauffman polynomial invariant F.

By using a graph diagram named a magnetic graph diagram, we construct a polynomial invariant for knots and links. We show that it is a generalization of both the HOMFLY and the Kauffman polynomials.

We construct an infinite series of invariants of Fiedler type (i.e. composed of oriented arrow diagrams arranged by elements of H₁(M³)) for multicomponent links in M³ = M² × R¹, M² orientable with π₁(M²) ≠ {1}.

Enhanced Yang–Baxter operators give rise to invariants of oriented links. We expand the enhancing method to generalized Yang–Baxter operators (gYB-operators). At present two examples of gYB-operators are known and recently three types of variations for one of these were discovered. We present the definition of enhanced generalized Yang–Baxter operators and show that all known examples of gYB-operators can be enhanced to give corresponding invariants of oriented links. Most of these invariants are specializations of the polynomial invariant P. Invariants from gYB-operators are multiplicative after a normalization.

In [Przytyski and Traczyk, Invariants of links of Conway type, Kobe J. Math.4 (1989) 115–139], Przytyski and Traczyk introduced an algebraic structure, called a Conway algebra, and constructed an invariant of oriented links, which is a generalization of the Homflypt polynomial invariant. On the other hand, in [Kauffman and Lambropoulou, New invariants of links and their state sum models, arXiv:1703.03655v2 [math.GT] 15 Mar 2017], Kauffman and Lambropoulou introduced new 4-variable invariants of oriented links, which are obtained by two computational steps: in the first step, we apply a skein relation on every mixed crossing to produce unions of unlinked knots. In the second step, we apply another skein relation on crossings of the unions of unlinked knots, which introduces a new variable. In this paper, we will introduce a generalization of the Conway algebra $Â$ with two binary operations and we construct an invariant valued in $Â$ by applying those two binary operations to mixed crossings and pure crossing, respectively. The 4-variable invariant of Kauffman and Lambropoulou with a specific condition is derived from the invariant valued in $Â$. Moreover, the generalized Conway algebra gives us an invariant of oriented links, which satisfies nonlinear skein relations.

A braided fusion category is said to have Property F if the associated braid group representations factor through a finite group. We verify integral metaplectic modular categories have property F by showing these categories are group-theoretical. For the special case of integral categories ${𝒞}$ with the fusion rules of $\mathrm{\text{SO}}{(8)}_2$ we determine the finite group $G$ for which $\mathrm{\text{Rep}}(D^{\omega }G)$ is braided equivalent to ${𝒵}({𝒞})$. In addition, we determine the associated classical link invariant, an evaluation of the 2-variable Kauffman polynomial at a point.

In this paper, we introduce an invariant of alternating knots and links (called here WRP), namely, a pair of integer polynomials associated with their two checkerboard planar graphs from their minimal diagram. We prove that the invariant is well-defined and give its values obtained from calculations for some knots in the tables. This invariant is strong enough to distinguish all knots in the tables with up to 10 crossings (including their mirror images). We compare the strength of the new invariant with classical invariants, including the three-variable Kauffman bracket.

We present the $CWR$ invariant, a new invariant for alternating links, which builds upon and generalizes the $WRP$ invariant. The $CWR$ invariant is an array of two-variable polynomials that provides a stronger invariant compared to the $WRP$ invariant. We compare the strength of our invariant with the classical HOMFLYPT, Kauffman $3$-variable, and Kauffman $2$-variable polynomials on specific knot examples. Additionally, we derive general recursive “skein” relations, and also specific formulas for the initial components of the $CWR$ invariant using weighted adjacency matrices of modified Tait graphs.

We introduce a generalization of spin models by dropping the symmetry condition. The partition function of a generalized spin model on a connected oriented link diagram is invariant under Reidemeister moves of type II and III, giving an invariant for oriented links.