Branched cyclic covers and finite type invariants

Abstract

This work identifies a class of moves on knots which translate to $m$-equivalences of the associated $p$-fold branched cyclic covers, for a fixed $m$ and any $p$ (with respect to the Goussarov-Habiro filtration.) These moves are applied to give a flexible (if specialised) construction of knots for which the Casson-Walker-Lescop invariant (for example) of their $p$-fold branched cyclic covers may be readily calculated, for any choice of $p$. In the second part of this paper, these operations are illustrated by some theorems concerning the relationship of knot invariants obtained from finite type three-manifold invariants, via the branched cyclic covering construction, with the finite type theory of knots.