Asymptotic behaviors in the homology of symmetric group and finite general linear group quandles

A quandle is an algebraic structure which attempts to generalize group conjugation. These structures have been studied extensively due to their connections with knot theory, algebraic combinatorics, and other fields. In this work, we approach the study of quandles from the perspective of the representation theory of categories. Namely, we look at collections of conjugacy classes of the symmetric groups and the finite general linear groups, and prove that they carry the structure of FI-quandles (resp. $\mathrm{VIC}(q)$-quandles). As applications, we prove statements about the homology of these quandles, and construct FI-module and $\mathrm{VIC}(q)$-module invariants of links.