No abstract

For ordinary knots in R 3 , there are no degree one Vassiliev invariants. For virtual knots, however, the space of degree one Vassiliev invariants is infinite dimensional. We introduce a sequence of three degree one Vassiliev invariants of virtual knots of increasing strength. We demonstrate that the strongest invariant is a universal Vassiliev invariant of degree one for virtual knots in the sense that any other degree one Vassiliev invariant can be recovered from it by a certain natural construction. To prove these results, we extend the based matrix invariant introduced by Turaev for virtual strings to the class of singular flat virtual knots with one doublepoint. Figure 3: Example of a knot and its Gauss diagramThere is a notion of equivalence for Gauss diagrams that corresponds to the equivalence of knot diagrams. In particular, there are combinatorial Reidemeister-type moves that relate Gauss diagrams representing the same knot, see [4].Every knot diagram represents an actual knot and can be described by a Gauss diagram, however there exist Gauss diagrams that do not correspond to ordinary knot diagrams. If we Here, the sum ranges over all non-virtual crossings d in K, while the value sign(d), the sign of d, is ±1, as defined in Figure 2. Proposition 4. S is a Vassiliev invariant of degree one.Proof. The proof of this proposition is similar to the proof of Proposition 2.Proposition 5. The smoothing invariant S is strictly stronger than the polynomial invariant p t .Proof. The absolute value of the intersection index |i(d)|, defined above, is an invariant of twocomponent flat virtual links. Thus, |i(d)| can be recovered from [ K d smooth ]. Hence, p t can be

Classical knots in R 3 can be represented by diagrams in the plane. These diagrams are formed by curves with a finite number of transverse crossings, where each crossing is decorated to indicate which strand of the knot passes over at that point. A pseudodiagram is a knot diagram that may be missing crossing information at some of its crossings. At these crossings, it is undetermined which strand passes over. Pseudodiagrams were first introduced by Ryo Hanaki in 2010. Here, we introduce the notion of a pseudoknot, i.e. an equivalence class of pseudodiagrams under an appropriate choice of Reidemeister moves. In order to begin a classification of pseudoknots, we introduce the concept of a weighted resolution set, or WeRe-set, an invariant of pseudoknots. We compute the WeRe-set for several pseudoknot families and discuss extensions of crossing number, homotopy, and chirality for pseudoknots.

In this paper, we introduce a new type of relation between knots called the descendant relation. One knot H is a descendant of another knot K if H can be obtained from a minimal crossing diagram of K by some number of crossing changes. We explore properties of the descendant relation and study how certain knots are related, paying particular attention to those knots, called fertile knots, that have a large number of descendants. Furthermore, we provide computational data related to various notions of knot fertility and propose several open questions for future exploration.

The Alexander biquandle of a virtual knot or link is a module over a 2-variable Laurent polynomial ring which is an invariant of virtual knots and links. The elementary ideals of this module are then invariants of virtual isotopy which determine both the generalized Alexander polynomial (also known as the Sawollek polynomial) for virtual knots and the classical Alexander polynomial for classical knots. For a fixed monomial ordering <, the Gröbner bases for these ideals are computable, comparable invariants which fully determine the elementary ideals and which generalize and unify the classical and generalized Alexander polynomials. We provide examples to illustrate the usefulness of these invariants and propose questions for future work.