Journal of Knot Theory and Its Ramifications

Given a virtual knot K, we introduce a new group-valued invariant VG_K called the virtual knot group, and we use the elementary ideals of VG_K to define invariants of K called the virtual Alexander invariants. For instance, associated to the zeroth ideal is a polynomial H_K(s, t, q) in three variables which we call the virtual Alexander polynomial, and we show that it is closely related to the generalized Alexander polynomial G_K(s, t) introduced by Sawollek; Kauffman and Radford; and Silver and Williams. We define a natural normalization of the virtual Alexander polynomial and show it satisfies a skein formula. We also introduce the twisted virtual Alexander polynomial associated to a virtual knot K and a representation ϱ : VG_K → GL_n(R), and we define a normalization of the twisted virtual Alexander polynomial. As applications we derive bounds on the virtual crossing numbers of virtual knots from the virtual Alexander polynomial and twisted virtual Alexander polynomial.

An n-crossing is a singular point in a projection of a link at which n strands cross such that each strand travels straight through the crossing. We introduce the notion of an übercrossing projection, a knot projection with a single n-crossing. Such a projection is necessarily composed of a collection of loops emanating from the crossing. We prove the surprising fact that all knots have a special type of übercrossing projection, which we call a petal projection, in which no loops contain any others. The rigidity of this form allows all the information about the knot to be concentrated in a permutation corresponding to the levels at which the strands lie within the crossing. These ideas give rise to two new invariants for a knot K: the übercrossing number ü(K), and petal number p(K). These are the least number of loops in any übercrossing or petal projection of K, respectively. We relate ü(K) and p(K) to other knot invariants, and compute p(K) for several classes of knots, including all knots of nine or fewer crossings.

This paper is devoted to the study of Hopf module structures on the spaces of framed chord diagrams and framed graphs. We also introduce a framed version of the chromatic polynomial and propose two methods to construct framed weight systems.

In this paper we give a simple proof of the equivalence between the rational link associated to the continued fraction [a₁, a₂,…,a_m], a_i ∈ ℕ, and the 2-bridge link of type p/q, where p/q is the rational number given by [a₁, a₂,…,a_m]. The known proof of this equivalence relies on the 2-fold cover of a link and the classification of the lens spaces. Our proof is elementary and combinatorial and follows the naïve approach of finding a set of movements to transform the rational link given by [a₁, a₂,…,a_m] into the 2-bridge link of type p/q.

In this paper we formalize a combinatorial object for describing link diagrams called a Planar Diagram Code (PD-Code). PD-codes are used by the KnotTheory Mathematica package developed by Bar-Natan et al. We present the set of PD-codes as a standalone object and discuss its relationship with link diagrams. We give an explicit algorithm for reconstructing a knot diagram on a surface from a PD-code. We also discuss the intrinsic symmetries of PD-codes (i.e. invertibility and chirality). The moves analogous to the Reidemeister moves are also explored, and we show that the given set of PD-codes modulo these combinatorial Reidemeister moves is equivalent to classical link theory.

We compute integer valued knot concordance invariants of a family of general pretzel knots if the invariants are equal to the negative values of signatures for alternating knots. Examples of such invariants are Rasmussen s-invariants and twice Ozsváth–Szabó knot Floer homology τ-invariants. We use the crossing change inequalities of Livingston and the fact that pretzel knots are almost alternating. As a consequence, for the family of pretzel knots given in this paper, s-invariants are twice τ-invariants.

When two boundary-parabolic representations of knot groups are given, we introduce the connected sum of these representations and show several natural properties including the unique factorization property. Furthermore, the complex volume of the connected sum is the sum of each complex volumes modulo iπ² and the twisted Alexander polynomial of the connected sum is the product of each polynomials with normalization.