Quantum invariant congruence for periodic links

Abstract

In this study, we introduce new criteria for a given link to detect the non-[Formula: see text]-periodicity for a prime [Formula: see text]. For this we use a congruence of the quantum [Formula: see text]-invariant of periodic links. Our result is naturally consistent with Traczyk’s criterion and Przytycki’s criterion, and can be regarded as a generalization of these classical criteria. We shall give computational results of all the (alternating and non-alternating) knots of crossing number [Formula: see text] for their non-[Formula: see text]-periodicity ([Formula: see text]). The computational results also show that our criteria has somewhat different nature with the classical criterion of Murasugi.