A characterization of the Γ-polynomials of knots with clasp number at most two

Abstract

It is known that every knot bounds a singular disk with only clasp singularities, which is called a clasp disk. The clasp number of a knot is the minimum number of clasp singularities among all clasp disks of the knot. It is known that the Conway polynomials of knots with clasp number at most two are characterized. In this paper, we focus on the common zeroth coefficient polynomial of both the HOMFLYPT and Kauffman polynomials, which is called the [Formula: see text]-polynomial. As a result, we characterize the [Formula: see text]-polynomials of knots with clasp number at most two. Moreover, it is known that there exist two homeomorphic classes of clasp disks with two clasp singularities, which are called types [Formula: see text] and [Formula: see text]. We show that there exist infinitely many knots which bound clasp disks of not doubled knots but both types [Formula: see text] and [Formula: see text], not type [Formula: see text] but type [Formula: see text], not type [Formula: see text] but type [Formula: see text], respectively.

An obstruction of Gordian distance one and cosmetic crossings for genus one knots

“…The zeroth coefficient polynomial of K = P (p, q, r) is computed from the skein relation, and given by p 0 K (v) = v p+q − v p+q+r+1 − v p+q+r−1 + v p+r + v q+r (see [Ta,Proposition 2.2 (i)], for example). Since p ≥ q > 1 > −1 > r p + q > p + q + r + 1 > p + q + r − 1 > p + r ≥ q + r If p 0 K (v) = f (v) 2 , then the coefficient of v p+q+r+1 should be even so p 0 K (v) is not the square of other polynomials.…”