An Alexander type invariant for doodles

Abstract

We construct an Alexander type invariant for oriented doodles from a deformation of the Tits representation of the twin group and from the Chebyshev polynomials of second kind. Similar to the Alexander polynomial, our invariant vanishes on unlinked doodles with more than one component. We also include values of our invariant on several doodles.

37 pages, many figures. v2: typos and table corrected, conjecture of v1 omitted