Narrow Results

We show that minimal genus Seifert surfaces for 2-bridge links can be isotoped to braided Seifert surfaces on minimal string braids. In particular, we can obtain a minimal string braid for a given 2-bridge link, where the word length of the braid thus obtained is shortest, among all braid presentations, in terms of the band generators of braids.

We discuss a conjecture of Jones on braid index and algebraic crossing number. We deform it to a stronger conjecture and show many evidences and some ways to approach the conjectures.

We showed the following in [8]: for any knot (i.e. one-component link) represented as a closed n-braid (n ≥ 3), there exists an infinite sequence of pairwise non-conjugate (n + 1)-braids representing the knot. Using the same technique, we construct an infinite sequence of pairwise non-conjugate 4-braids representing the same knot of braid index 4. Consequently, we have that M. Hirasawa's candidates are actually such infinite sequences.