Milnor’s work and knot polynomials. (English) Zbl 0833.57001

Jones, V. F. R.

Goldberg, Lisa R. (ed.) et al., Topological methods in modern mathematics. Proceedings of a symposium in honor of John Milnor’s sixtieth birthday, held at the State University of New York at Stony Brook, USA, June 14-June 21, 1991. Houston, TX: Publish or Perish, Inc. 195-202 (1993).

In this interesting paper, the author discusses three topics that J. Milnor studied in later 1950’s. These are: the unknotting number \(u(K)\) of a torus knot \(K\), the bridge number via crookedness and link homotopy. The so-called Milnor conjecture that \(u(K)=\) genus of \(K\) for a torus knot, has been solved quite recently by P. Kronheimer and T. Mrowka after this paper was written. All these topics are reviewed from the recently developed mechanism using Jones polynomial and its generalization. The author suggests a new approach to these problems.
For the entire collection see [Zbl 0780.00031].

Reviewer: K.Murasugi (Toronto)

MSC:

57M25	Knots and links in the \(3\)-sphere (MSC2010)
57-02	Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes

Keywords:

unknotting number; torus knot; bridge number; link homotopy; Jones polynomial