Title : Enumerating the prime knots and links by a canonical order II
(a joint work with Akio Kawauchi (Osaka City University))
Speaker : Ikuo Tayama (Osaka City University)
Abstract :

A well-order (called a {\it canonical order}) was introduced in the set of links by A. Kawauchi [K] (see also A. Kawauchi and I. Tayama [KT]). This well-order also naturally induces a well-order in the set of closed connected orientable $3$-manifolds and suggests a method for enumerating the prime links and the $3$-manifolds. We assign to every link a lattice point whose length is equal to the minimal crossing number on closed braid forms of the link and we call the number the {\it length} of the link. We note that a link $L$ is smaller than a link $L'$ in the canonical order if the length of $L$ is smaller than that of $L'$, and for any natural number $n$ there are only finitely many links with lengths up to $n$. In this talk, we give a way to enumerate the prime links by the canonical order and show a table of the prime links with lengths up to 10. \smallskip \begin{center} {\bf References} \end{center} \smallskip [K] A. Kawauchi, A tabulation of 3-manifolds via Dehn surgery, Boletin de la Sociedad Matematica Mexicana(to appear). [KI] A. Kawauchi and I. Tayama, Enumerating the prime knots and links by a canonical order, Proc. 1st East Asian School of Knots, Links, and Related Topics, 2004 (to appear).

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