Two dimensional arrays for Alexander polynomials of torus knots

Abstract

Given a pair $p,q$ of relative prime positive integers, we have uniquely determined positive integers $x,y,u$ and $v$ such that $vx-uy=1$, $p=x+y$ and $q=u+v$. Using this property, we show that $$ \sum_{1\leq i\leq x, 1\leq j \leq v} t^{(i-1)q+(j-1)p}~ -\sum_{1\leq k\leq y, 1\leq l \leq u} t^{1+(k-1)q+(l-1)p}$$ is the Alexander polynomial $\Delta_{p,q}(t)$ of a torus knot $t(p,q)$. Hence the number $N_{p,q}$ of non-zero terms of $\Delta_{p,q}(t)$ is equal to $vx+uy=2vx-1$. Owing to well known results in knot Floer homology theory, our expanding formula of the Alexander polynomial of a torus knot provides a method of algorithmically determining the total rank of its knot Floer homology or equivalently the complexity of its (1,1)-diagram. In particular we prove (see Corollary~\ref{c03}); Let $q$ be a positive integer$>1$ and let $k$ be a positive integer. Then we have $$\begin{array}{lrrl} (1)&N_{kq+1,q}&=&2k(q-1)+1\\ (2)&N_{kq+q-1,q}&=&2(k+1)(q-1)-1\\ (3)&N_{kq+2,q}&=&\frac{1}{2}k(q^2-1)+q\\ (4)&N_{kq+q-2,q}&=&\frac{1}{2}(k+1)(q^2-1)-q \end{array}$$ where we further assume $q$ is odd in formula $(3)$ and $(4)$. Consequently we confirm that the complexities of (1,1)-diagrams of torus knots of type $t(kq+2,q)$ and $t(kq+q-2,q)$ in \cite{KK} agree with $N_{kq+2,q}$ and $N_{kq+q-2,q}$ respectively.