Unknotting singular surface braids by crossing changes

MASAHIDE IWAKIRI

Abstract

A surface braid was introduced by O. Viro [8] and has extensively studied by S. Kamada [6]. L. Rudolph introduced a similar notion earlier in [7]. By a singular surface braid, we mean an immersed (closed) surface braid. We call a transverse double point of a singular surface braid a crossing point. By a crossing change, we mean an operation for a singular surface braid S inserting a pair of positive and negative crossing points along a chord that is a straight segment connecting adjacent sheets of S (cf. [2, 3]). In this talk, we present an unknotting theorem (Theorem 1), which is given when they have no branch points by Kamada [2].

Theorem 1. Any singular surface braid can be transformed to an unknotted one by crossing changes and its inverse operations.

C. A. Giller [1, 5] proved that such an unknotting theorem for surfaces in Euclidean 4-space. As a consequence, we have this Giller’s unknotting theorem.

Corollary 2 ([1, 3]). Any surface in 4-space can be transformed to an unknotted one by crossing changes and its inverse operations.

References

[1] C. A. Giller, Towards a classical knot theory for surfaces in R^{4}, Illinois J. Math. 26 (1982), 591–631.

[2] S. Kamada, Crossing changes for singular 2-dimensional braids without branch points, Kobe J. Math., 13 (1996), 177-182.

[3] S. Kamada, Surfaces in 4-space: a view of normal forms and braidings, Lectures at KNOTS 96 (ed S.Suzuki) (1997), World Sci. Publishing Co. 39-71.

[4] S. Kamada, Unknotting immersed surface-links and singular 2-dimension braids by 1-handle surgeries, Osaka J. Math. 36 (1999) 33-49.

[5] S. Kamada, Vanishing of a certain kind of Vassiliev invariants of 2-knots, Proc. Amer. Math. Soc. 127 (1999) 3421-3426.

[6] S. Kamada, Braid and knot theory in dimension four, Math. Surveys Monogr. 95, Amer. Math. Soc., 2002.

[7] L. Rudolph, Braided surfaces and Seifert ribbons for closed braids, Comment. Math. Helv, 58 (1983) 1-37.

[8] O. Ya. Viro, Lecture given at Osaka City University, September, 1990.

Department of Mathematics Hiroshima University, Hirohsima 739-8526, JAPAN