Detecting loops of singular long knots with polynomial complexity

Abstract: Let l be a loop of long knots with at least one ordinary double point and which is given by a 1-parameter family of diagrams in the moduli space of singular long knots. We assume that the Alexander polynomial for singular knots is non trivial for the orientation preserving smoothing of at least one of the double points of the knot. We construct a 1-cocycle in the moduli space and use it, together with the efficient calculation of Alexander polynomials for singular knots by Schut and van der Veen, in order to determine the homology class represented by the loop l in the moduli space. Surprisingly, this can be done with polynomial complexity with respect to the number of Reidemeister moves in the loop l and with respect to the maximal number of crossings of the knots in the loop.

This is research done with our working group of Louis Kauffman, Sofia Lambropoulou, Tumpa Mahato, Kasturi Barkataki and Hamdi Kayaslan.

mathematical physicsalgebraic topologycombinatoricsgeometric topologyrings and algebrasrepresentation theory

Audience: researchers in the topic

Knots, graphs and groups

Series comments: Zoom - Meeting ID: 818 6674 5751 Passcode: 141592

Link: us02web.zoom.us/j/81866745751?pwd=bEFqUUlZM1hVV0tvN0xWdXRsV2pnQT09

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