A lifting defect for oriented $t_3$ and $\bar t_6$ moves

Abstract

We study oriented $t_3$- and $\bar t_6$-moves through the problem of lifting Fox $3$-colorings to colorings by a nine-element Alexander quandle. We define a nonnegative integer lifting defect $δ$ for oriented links and prove that it is invariant under these moves. For knots, the defect equals the rank of an alternating bilinear form obtained from a Seifert matrix, so it is even. We also prove that it is additive under connected sum. Let $K_s$ be the connected sum of $s\geq 1$ copies of $9_{46}$. Then $δ(K_s)=2s$, so these knots represent infinitely many distinct equivalence classes under the oriented moves, and none is equivalent to an unlink. After forgetting orientation, each $K_s$ can be transformed into an unlink with $2s+1$ components by $3s$ ordinary $3$-moves.