Renormalizations of the Kontsevich integral and their behavior under band sum moves.

Abstract

We generalize the definition of the framed Kontsevich integral initially presented in [LM1]. We study the behavior of the renormalized framed Kontsevich integral Z[hat]_f under band sum moves and show that it can be further renormalized into some invariant Z[widetilde]_f that is well-behaved under moves for which link components of interest are locally put on top of each other. Originally, Le, Murakami and Ohtsuki ([LM5], [LM6]) showed that another choice of normalization is better suited for moves for which link components involved in the band sum move are put side by side. We show the choice of renormalization leads to essentially the same invariant and that the use of one renormalization or the other is just a matter of preference depending on whether one decides to have a horizontal or a vertical band sum. Much of the work on Z[widetilde]_f relies on using the tangle chord diagrams version of Z[hat]_f ([ChDu]). This leads us to introducing a matrix representation of tangle chord diagrams, where each chord is represented by a matrix, and tangle chord diagrams of degree $m$ are represented by stacks of m matrices, one for each chord making up the diagram. We show matrix congruences for some appropriately chosen matrices implement on the modified Kontsevich integral Z[widetilde]_f the band sum move on links. We show how Z[widetilde]_f in matrix notation behaves under the Reidemeister moves and under orientation changes. We show that for a link L in plat position, Z_f(L) in book notation is enough to recover its expression in terms of chord diagrams. We elucidate the relation between Z[check]_f and Z[widetilde]_f and show the quotienting procedure to produce 3-manifold invariants from those as introduced in [LM5] is blind to the choice of normalization, and thus any choice of normalization leads to a 3-manifold invariant.