Colored finite type invariants and a multi-variable analogue of the Conway polynomial

Details Colored finite type invariants and a multi-variable analogue of the Conway polynomial Colored finite type invariants and a multi-variable analogue of the Conway polynomial

2003-11-29

arxiv.org/abs/math/0312007v2

Mathematics

Geometric Topology

23 pages. The second half of the title and the first half of the abstract are new, reflecting new material added in v2

Scientific paper

The multi-variable Alexander polynomial (in the form of Conway's potential function), when stripped of a redundant summand, is shown to be of the form \Nabla_L(x_1-x_1^{-1},..,x_m-x_m^{-1}) for some polynomial \Nabla_L over Z. The Conway polynomial \nabla_L(z) coincides with z\Nabla_L(z,..,z). The coefficients of \Nabla_L and of the power series \Nabla^*_L:=\Nabla_L/(\Nabla_{K_1}...\Nabla_{K_m}), where K_i denote the components of L, are finite type invariants in the sense of Kirk-Livingston. When m=2, they are integral liftings of Milnor's invariants \bar\mu(1..12..2) of even length, including, in the case of \Nabla^*_L, Cochran's derived invariants \beta^k. The coefficients of \Nabla_L and \Nabla^*_L are closely related to certain Q-valued invariants of (genuine) finite type, among which we find alternative extensions \hat\beta^k of \beta^k to the case lk\ne 0, such that 2\hat\beta^1/lk^2 is the Casson-Walker invariant of the Q-homology sphere obtained by 0-surgery on the link components. Each coefficient of \Nabla^*_L (hence of \nabla^*_L:=\nabla_L/(\nabla_{K_1}...\nabla_{K_m})) is invariant under TOP isotopy and under sufficiently close C^0-approximation, and can be extended, preserving these properties, to all topological links. The same holds for H^*_L and F^*_L, where H_L and F_L are certain exponential parameterizations of the two-variable HOMFLY and Kauffman polynomials. Next, we show that no difference between PL isotopy and TOP isotopy (as equivalence relations on PL links in S^3) can be detected by finite type invariants. These are corollaries of the fact that any type k invariant (genuine or Kirk-Livingston), well-defined up to PL isotopy, assumes same values on k-quasi-isotopic links.

Affiliated with

Also associated with

No associations

Rating

Colored finite type invariants and a multi-variable analogue of the Conway polynomial does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Colored finite type invariants and a multi-variable analogue of the Conway polynomial, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Colored finite type invariants and a multi-variable analogue of the Conway polynomial will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-643942