Classical and Virtual Pseudodiagram Theory and New Bounds on Unknotting Numbers and Genus

Details Classical and Virtual Pseudodiagram Theory and New Bounds on Unknotting Numbers and Genus Classical and Virtual Pseudodiagram Theory and New Bounds on Unknotting Numbers and Genus

2009-08-14

arxiv.org/abs/0908.1981v2

J. Knot Theory Ramifications, Vol. 20, No. 4 (2011) pp. 625-650

Mathematics

Geometric Topology

25 pages, 18 figures, 1 table

Scientific paper

10.1142/S0218216511009388

A pseudodiagram is a diagram of a knot with some crossing information missing. We review and expand the theory of pseudodiagrams introduced by R. Hanaki. We then extend this theory to the realm of virtual knots, a generalization of knots. In particular, we investigate how much crossing information must be known to conclude that a diagram is a diagram of the unknot (the trivializing number). We also consider how much information is necessary to identify a non-trivial knot, a classical knot, or a non-classical knot. We then apply pseudodiagram theory to develop new upper bounds on unknotting number, virtual unknotting number, and genus.

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