A Sequence of Degree One Vassiliev Invariants for Virtual Knots

Details A Sequence of Degree One Vassiliev Invariants for Virtual Knots A Sequence of Degree One Vassiliev Invariants for Virtual Knots

2008-03-05

arxiv.org/abs/0803.0754v3

Journal of Knot Theory and its Ramifications 19, 4 (2010), pp. 461--487

Mathematics

Geometric Topology

26 pages, 19 figures, modified example and updated references

Scientific paper

10.1142/10.1142/S021821651000791

For ordinary knots in R3, 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 virtual knots with one double-point.

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