Narrow Results

In this paper, we express the Seifert matrix of a periodic link which is presented as the closure of a 4-tangle with some extra restrictions, in terms of the Seifert matrix of the quotient link. As a result, we give formulae for the Alexander polynomial and the determinant of such a periodic link.

We show that for any positive integers $a,b,m,$ and $n$, the Alexander polynomial of the $(am,bn)$-Turk’s head link is divisible by that of the $(m,n)$-Turk’s head link.

In this paper, we provide two new congruences of the generalized Alexander polynomial $Z_L$ for periodic virtual links $L$. We use the Yang–Baxter state model of $Z_L$ introduced by Kauffman and Radford.

In this study, we introduce new criteria for a given link to detect the non-$p$-periodicity for a prime $p\ge 5$. For this we use a congruence of the quantum ${𝔰𝔩}_N$-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 $\le 16$ for their non-$p$-periodicity ($p\ge 5$). The computational results also show that our criteria has somewhat different nature with the classical criterion of Murasugi.