Twist of Knots and the Q-polynomials
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Abstract
The last two authors ([16]) gave solutions for the problem whether a higher derivative of the Conway, Alexander and Jones polynomial at a point is a Vassiliev invariant or not, by using Birman and Lin’s result ([2]). For the Q-polynomial it is known that the nth derivative Q(n )K (a) of theQ polynomial QK (x) of a knot K at a is not a Vassiliev invariant if a 6= 1 ; 2 ([16], [39]). A sequence fK ig1i=0 of knots is called a twist sequence if they differ in a local part of two strands in whichK i+1 is obtained from K i by adding a full twist for each i. The local transform of two parallel strands with parallel orientation to the k-half twist of the two strands is called the tk-move.In this paper we show that, for any positive integer n,Q(n )K (1) is not a Vassiliev invariant and Q(n )K ( 2) is not a Vassiliev invariant of degree< 2n, by using R. Trapp’sresult ([41]) on twist sequences of knots. Also by using higher derivatives Q(n )K ( 2) of the Q polynomial, we give some criterions to detect whether a knot K can be transformed to a knot K0 by finitely many t2k-moves, and if so, we give some results on the number of t2k-moves necessary in the transformation.
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