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G&T Monographs |
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The nonuniqueness of Chekanov polynomials of Legendrian
knots
Paul Melvin and Sumana Shrestha
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Geometry & Topology 9 (2005)
1221–1252
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Abstract
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Examples are given of prime Legendrian knots in the standard contact 3–space that
have arbitrarily many distinct Chekanov polynomials, refuting a conjecture of Lenny
Ng. These are constructed using a new “Legendrian tangle replacement” technique.
This technique is then used to show that the phenomenon of multiple Chekanov
polynomials is in fact quite common. Finally, building on unpublished work of Yufa
and Branson, a tabulation is given of Legendrian fronts, along with their Chekanov
polynomials, representing maximal Thurston–Bennequin Legendrian knots for each
knot type of nine or fewer crossings. These knots are paired so that the front for the
mirror of any knot is obtained in a standard way by rotating the front for the
knot.
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Keywords
Legendrian knots, contact homology,
Chekanov polynomials
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Mathematical Subject Classification
Primary: 57R17
Secondary: 53D12, 57M25
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Publication
Received: 10 November 2004
Revised: 3 December 2004
Accepted: 4 July 2005
Published: 24 July 2005
Proposed: Yasha Eliashberg
Seconded: Robion Kirby, Joan Birman
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