The spiral index of knots

Details The spiral index of knots The spiral index of knots

2009-03-03

arxiv.org/abs/0903.0393v1

Math. Proc. of the Cambridge Philosophical Society, Vol. 149, Issue 2, 297-315 (2010)

Mathematics

Geometric Topology

19 pages, 14 figures

Scientific paper

10.1017/S0305004110000241

Two new invariants that are closely related to Milnor's curvature-torsion invariant are introduced. The first, the spiral index of a knot, captures the minimum number of maxima among all knot projections that are free of inflection points. This invariant is closely related to both the bridge and braid index of the knot. The second, the projective superbridge index, provides a method of counting the greatest number of maxima that occur in a given knot projection. In addition to investigating how these invariants are related to the classical invariants, we utilize them to determine all knots with curvature-torsion invariant equal to 6 pi.

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