A new criterion for knots with free periods

Abstract

Abstract. Let p ≥ 2 and q = 0 an integer. A knot K in the three-sphere is said to be a (p, q)-lens knot if and only if it covers a link in the lens space L(p, q). In this paper, we use the second coefficient of the

An explicit relation between knot groups in lens spaces and those in $S^3$

“…Then there exists an integer q and a knot K ′ in L(p, q) such that π −1 (K ′ ) is isotopic to the torus knot T m,n or its mirror image if and only if gcd(mn, p) = 1. Furthermore, Chbili [5] determined the possible values of the above q. We also observe it in Proposition 3.4 via Corollary 1.…”

Lift in the 3-sphere of knots and links in lens spaces

“…See Figure 10 for an example. In this case, we say that B represents L. [Ch2], that generalizes a result of [H] for torus knots. Remember that the torus link T n,m ⊂ S 3 is the closure of the braid…”

“…The lift in S 3 of a link in L(p, q) is exactly a (p, q)-lens link of Chbili [Ch2], and hence a freely periodic link in the 3-sphere [H]. Our question can be re-phrased: "Are there links in S 3 that are freely periodic with respect to two different (p, q)-periodic transformations?…”

Graph polynomials and symmetries

“…The Tutte polynomial is also related to certain specialization of the HOMFLYPT polynomial as it was proved in [4]. The latter is known to be a good witness of link symmetries, see [1,2,7,11,12,14]. The purpose of this paper is to investigate whether the way the Tutte polynomial interacts with graph symmetries extends to other graph polynomials.…”