Computations of parabolic character schemes of knots

Abstract

We compute parabolic ${\mathrm{SL}}_2(\mathbb{C})$-character schemes of knots using the parabolic quandle. To this end, we introduce sign-refined arc-colorings and show that their sign data encode the obstruction classes of the induced parabolic representations. We also establish a correspondence between the schemes defined by sign-refined arc-colorings and the parabolic character scheme. This yields a practical diagrammatic method for computing complete lists of parabolic characters, together with their multiplicities and obstruction classes. Using this method, we verify a conjecture of Bénard and Detcherry for all small knots with at most $12$ crossings.

21 pages. Some of the computational data were previously presented in arXiv:2204.00319