Geometry & Topology

The nonuniqueness of Chekanov polynomials of Legendrian knots

Paul Melvin and Sumana Shrestha

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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 Finally, building on unpublished work of Yufa and Branson, a tabulation is given of Legendrian fronts, along with their Chekanov polynomials, representing maximal Thurston–Bennequin Legendrian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.

Article information

Geom. Topol., Volume 9, Number 3 (2005), 1221-1252.

Received: 10 November 2004
Revised: 3 December 2004
Accepted: 4 July 2005
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57R17: Symplectic and contact topology
Secondary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45} 53D12: Lagrangian submanifolds; Maslov index

Legendrian knots contact homology Chekanov polynomials


Melvin, Paul; Shrestha, Sumana. The nonuniqueness of Chekanov polynomials of Legendrian knots. Geom. Topol. 9 (2005), no. 3, 1221--1252. doi:10.2140/gt.2005.9.1221.

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