A Sequence of Degree One Vassiliev Invariants for Virtual Knots

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

Index polynomials for virtual tangles

“…Ever since its introduction in [Hen10] and [Tur08], index polynomial invariants of virtual knots have been a very popular topic, see for example [Kau13], [FK13], [ILL10]. Because of their construction, many of these turn out to be Vassiliev invariants of order one for virtual knots, a notion first introduced in [Kau99] as a natural generalization of finite-type invariants of classical knots.…”

The multi-variable Affine Index Polynomial

“…21 is a link, that example does not work for the AIP for virtual knots; instead, we've provided the example in Fig. 22, which already appears as an example in [Hen10]. Resolving the double point one way yields the unknot (whose polynomial is zero), while the other resolution has an AIP of t 2 + t −2 − 2.…”

Homotopy invariants of Gauss words

“…However, using information in the projection it should be possible to define stronger invariants based on z. Indeed, as we mentioned in the introduction, Henrich's smoothing invariant [4] is an example of such an invariant in the case of virtual knots.…”

“…This invariant was inspired by Henrich's smoothing invariant for virtual knots defined in [4]. In fact, our invariant can be viewed as a version of Henrich's invariant, weakened sufficiently to remain invariant under homotopy of Gauss words.…”