Mathematical Proceedings of the Cambridge Philosophical Society

Research Article

Brunnian links, claspers and Goussarov–Vassiliev finite type invariants

KAZUO HABIROa1

a1 Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan. e-mail: habiro@kurims.kyoto-u.ac.jp

Abstract

Goussarov and the author independently proved that two knots in S3 have the same values of finite type invariants of degree <n if and only if they are Cn-equivalent, which means that they are equivalent up to modification by a kind of geometric commutator of class n. This property does not generalize to links with more than one component.

In this paper, we study the case of Brunnian links, which are links whose proper sublinks are trivial. We prove that if n ≥ 1, then an (n+1)-component Brunnian link L is Cn-equivalent to an unlink. We also prove that if n ≥ 2, then L can not be distinguished from an unlink by any Goussarov–Vassiliev finite type invariant of degree <2n.

(Received January 04 2006)

(Revised May 30 2006)

Footnotes

† Partially supported by the Japan Society for the Promotion of Science, Grant-in-Aid for Young Scientists (B) 16740033.

Dedicated to the memory of Mikhail Goussarov