Twins / Doodles

This is a collection of doodles and their values under an invariant we construct from a deformation of the Tits representation of the twin group and the Chebyshev polynomials of second kind. This is part of the paper An Alexander type invariant for doodles (arXiv:2005.06290), with Bruno Cisneros, Marcelo Flores and Jesús Juyumaya.

The first line following the picture is the number of crossings of the minimal representative and the superscript means the number of components. The second line is the twin representation of the doodle, where the number k denotes the k-th standard generator of the twin group. The last line codified the polynomial invariant, the first number in curly brackets denotes the half of the maximum degree of the polynomial and the next sequence in parenthesis denotes the coefficients, from higher to lower degree. For instance, {7}(1,2,-1,-2,1) denotes the polynomial t^{7} + 2t^{6} - t^{5} - 2t^{4} + t^{3}.

This is a collection of generators of the pure twin group in 6 strands. This was the beginning of the paper Planar pure braids on six strands (arXiv:1905.08326), with Jacob Mostovoy.