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Finite type invariants based on the band-pass and doubled delta moves

James Conant·Jacob Mostovoy·Ted Stanford
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Abstract

We study generalizations of finite-type knot invariants obtained by replacing the crossing change in the Vassiliev skein relation by some other local move, analyzing in detail the band-pass and doubled-delta moves. Using braid-theoretic techniques, we show that, for a large class of local moves, generalized Goussarov's n-equivalence classes of knots form groups under connected sum. (Similar results, but with a different approach, have been obtained before by Taniyama and Yasuhara.) It turns out that primitive band-pass finite-type invariants essentially coincide with standard primitive finite-type invariants, but things are more interesting for the doubled-delta move. The complete degree 0 doubled-delta invariant is the $S$-equivalence class of the knot. In this context, we generalize a result of Murakami and Ohtsuki to show that the only primitive Vassiliev invariants of S-equivalence taking values in an abelian group with no 2-torsion arise from the Alexander-Conway polynomial. We start analyzing degree one doubled-delta invariants by considering which Vassiliev invariants are of doubled-delta degree one, finding that there is exactly one such invariant in each odd Vassiliev degree, and at most one (which is Z/2-valued) in each even Vassiliev degree. Analyzing higher doubled-delta degrees, we observe that the Euler degree n+1. part of Garoufalidis and Kricker's rational lift of the Kontsevich integral is a doubled-delta degree $2n$ invariant.

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James Conant, Jacob Mostovoy, Ted Stanford. 2009-01-14. Finite type invariants based on the band-pass and doubled delta moves. https://arxiv.org/abs/math/0511189

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