Longitudes in $\mathrm{SL}_2$ representations of link groups and Milnor–Witt $K_2$-groups of fields

Abstract

We describe an arithmetic $K_2$-valued invariant for longitudes of a link $L\subset {ℝ}^3$, obtained from an ${SL}_2$ representation of the link group. Furthermore, we show a nontriviality on the elements, and compute the elements for some links. As an application, we develop a method for computing longitudes in ${\widetilde{SL}}_2^{top}\left(ℝ\right)$ representations for link groups, where ${\widetilde{SL}}_2^{top}\left(ℝ\right)$ is the universal covering group of ${SL}_2\left(ℝ\right)$.