Mathematics – Geometric Topology
Scientific paper
2010-02-02
Mathematics
Geometric Topology
36 pages, v2--Minor changes to exposition; added Theorem 12; Corrected typo in table on page 34
Scientific paper
The celebrated theorem of Goussarov states that all finite-type (Vassiliev-Goussarov) invariants of classical knots can be expressed in terms of Polyak-Viro combinatorial formulae. These formulae intrinsically use non-realizable Gauss diagrams and virtual knots. Some of these formulae can be naturally extended to virtual knots; however, the class of finite-type invariants of virtual knots obtained by using these formulae (so-called Goussarov-Polyak-Viro finite-type invariants) is very small. Kauffman gave a more natural notion of finite-type invariants, which, however, turned out to be quite complicated: even invariants of order zero form an infinite-dimensional space. Recently, the second named author introduced the notion of {\em parity} which turned out to be extremely useful for many purposes in virtual knot theory and low-dimensional topology; in particular, they turned out to be useful for constructing invariants of {\em free knots}, the latter being very close to the notion of order 0 invariants. In the present paper we use the concept of parity to enlarge the notion of Goussarov-Polyak-Viro combinatorial formulae and provide explicit formulae for these invariants. Not all of the new invariants are of GPV finite-type, but they all are of Kauffman finite-type. Also, we establish some relations with the with the standard GPV formulae.
Chrisman Micah
Manturov Vassily Olegovich
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