Fiber quadrisecants in knot isotopies

Details Fiber quadrisecants in knot isotopies Fiber quadrisecants in knot isotopies

2007-01-30

arxiv.org/abs/math/0701878v1

Mathematics

Geometric Topology

12 pages, 8 figures

Scientific paper

Fix a straight line L in Euclidean 3-space and consider the fibration of the complement of L by half-planes. A generic knot K in the complement of L has neither fiber quadrisecants nor fiber extreme secants such that K touches the corresponding half-plane at 2 points. Both types of secants occur in generic isotopies of knots. We give lower bounds for the number of these fiber secants in all isotopies connecting given isotopic knots. The bounds are expressed in terms of invariants calculable in linear time with respect to the number of crossings.

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