The MVA and Polynomiality
August-12-10
4:19 PM
MVA := The Multi-Variable Alexander polynomial.
SVA := The Single Variable Alexander polynomial.
Everything is measured with the number of crossings as the complexity parameter.
Is the MVA computable in polynomial time? Every evaluation thereof at fixed rational numbers is, and so is every reduction thereof to a fixed number of variables. But is the whole thing poly-time computable? (The SVA is poly-time computable no questions asked).
Main Question: Is there an extension of the MVA to tangles (in the circuit- or planar algebra sense) which composes well and that can be computed in poly-time?
Same question, for the SVA. (Known for braids - this is Burau).
Same question, for the MVA evaluated at specific rational numbers.
I think I can do better than Turaev-Cimasoni, Kirk-Livingston-Wang, etc., in the sense that I have a poly-time invariant of open tangles (only in specific evaluations) which is well behaved under compositions and from which the invariants of closures can be computed. Yet I don't understand algorithmically the last step of "closure".