Title : On the $z$-degree of the Kauffman polynomials via a tangle decomposition
Speaker : Yongju Bae (Kyungpook National University)
Abstract :

In 1987, M.E. Kidwell produced an upper bound on the degree of the Dubrovnik polynomial $D_L(\lambda,z)$ in terms of the crossing number and the length of the longest bridge in a link diagram. Indeed, $\max\deg_z D_L(\lambda,z)\leq c(D)-b(D),$ where $D$ is a diagram of a link $L$ with $c(D)$ crossings and $b(D)$ is the length of a longest bridge in $D.$ It is natural to ask the following question; \noindent{\bf Question.} {\it How the maximal $z$-degree of $D_L(\lambda,z)$ is affected by a set of bridges $B_1,\cdots,B_n$ in a link diagram $D$?} It is trivial that if $D$ is a connected sum of $D_1, D_2,\cdots,D_n$ and if $b(D_i)$ is the length of a longest bridge in $D_i,$ then $\max\deg_z D_D(\lambda,z)\leq \sum_{i=1}^n(c(D_i)-b(D_i)),$ where $c(D_i)$ denote the number of crossings in $D_i$. In 2001, Kidwell and Stanford showed the following which is a partial solution of the question: \begin{prop} Let $D$ be a link diagram written as a wiring diagram with $n$ $4$-tangles $\{T_i\}_{i=1}^n$. Let $c(T_i)$ be the number of crossings in $T_i$ and $b(T_i)$ the length of a longest bridge in $T_i.$ $$\max\deg_z D_D(\lambda,z)\leq \sum_{i=1}^n(c(T_i)-b(T_i))+(n-1).$$ \end{prop} In this talk, we will generalize the results of Kidwell and Stanford for {\em any} tangles. \begin{thm} Let $D$ be a link diagram written as a wiring diagram with $n$ $2k_i$-tangles $\{T_i\}_{i=1}^n$. Let $c(T_i)$ be the number of crossings in $T_i$ and $b(T_i)$ the length of a longest bridge in $T_i.$ Then $$\max\deg_z D_D(\lambda,z)(x)\leq \sum_{i=1}^n(c(T_i)-b(T_i))+\sum_{i=2}^n\frac{k_i(k_i-1)}{2}.$$ \end{thm}

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