Title : Twist of knots and the $Q$-polynomial
(a jointwork with Myeong--Ju Jeong and Younhee Choi)
Speaker : Chan-Young Park (Kyungpook National University)
Abstract :

For the $Q$-polynomial it is known that the $n$--th derivative $Q_K^{(n)}(a)$ of the $Q$--polynomial $Q_K(x)$ of a knot $K$ at $a$ is not a Vassiliev invariant if $a \neq 1, -2$. The local transformation of two parallel strands with parallel orientation to the $k$--half twist of the two strands is called the $t_k$--move. In this talk we show that, for any positive integer $n$, $Q_K^{(n)}(1)$ is not a Vassiliev invariant and $Q_K^{(n)}(-2)$ is not a Vassiliev invariant of degree $< 2n$, by using R. Trapp's result %(\cite{Tr}) on twist sequences of knots. Also by using higher derivatives $Q_K^{(n)}(-2)$ of the $Q$--polynomial, we give some criterions to detect whether a knot $K$ can be transformed to a knot $K'$ by finitely many $t_{2k}$--moves, and if so, we give some results on the number of $t_{2k}$--moves necessary in the transformation. This is a jointwork with Dr. Myeong--Ju Jeong and Ms. Younhee Choi.

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