Equivalence of symmetric union diagrams

Details Equivalence of symmetric union diagrams Equivalence of symmetric union diagrams

2007-05-31

arxiv.org/abs/0705.4578v2

J. Knot Theory Ramifications 16 (2007) 879-898

Mathematics

Geometric Topology

19 pages, 20 figures; v2 corrected signs in section 3

Scientific paper

10.1142/S0218216507005555

Motivated by the study of ribbon knots we explore symmetric unions, a beautiful construction introduced by Kinoshita and Terasaka 50 years ago. It is easy to see that every symmetric union represents a ribbon knot, but the converse is still an open problem. Besides existence it is natural to consider the question of uniqueness. In order to attack this question we extend the usual Reidemeister moves to a family of moves respecting the symmetry, and consider the symmetric equivalence thus generated. This notion being in place, we discuss several situations in which a knot can have essentially distinct symmetric union representations. We exhibit an infinite family of ribbon two-bridge knots each of which allows two different symmetric union representations.

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