On periodic knots

Square numbers, spanning trees and invariants of achiral knots

“…As in [4], K 1 and K have the same Alexander polynomial, so in particular, the same determinant. 42 and 10 125 , the congruence modulo 4 is violated, so that, as remarked on several other places, the signature works as well.) Remark 4.6.…”

“…Whenever ∆(−1) < 0, we have σ ≡ 2 mod 4, so in particular σ = 0, and the knot cannot be achiral. This argument applies for example (but not only) for the knot 9 42 . Note that the normalization ∆(1/t) = ∆(t) implies ∆(1) ≡ ∆(−1) mod 4, so that if ∆(1) = 1, the property ∆(−1) < 0 is also equivalent to det(…”

“…with K =!K) but self-conjugate polynomials is 9 42 in the tables of [49, appendix]. Let us recall a paradox appearing argument showing only using ∆ 9 42 = ∆ !9 42 that 9 42 =!9 42 . For this (and also for the purpose of the main work in this paper) we will be concerned with the determinant.…”

Determinant Formulae of Matrices with Certain Symmetry and Its Applications

“…One of an important concern of knot theory is to find the relationship between periodic links and their factor links, see [17,18]. In 2011, the authors expressed 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 quotient link in [2].…”

Homologically trivial actions on cyclic coverings of knots

“…i) The Alexander polynomial of a periodic knot (see Section 2 for precise definitions and statements) satisfies the so-called Murasugi conditions [21]. These conditions can be explicitly computed and have been used to test periods of knots [4].…”