Thurston–Bennequin number
In the mathematical theory of knots, the Thurston–Bennequin number, or Bennequin number, of a front diagram of a Legendrian knot is defined as the writhe of the diagram minus the number of right cusps[1]. It is named after William Thurston and Daniel Bennequin.
The Thurston-Bennequin number of a knot is commonly denoted by . The maximal Thurston–Bennequin number, , over all Legendrian representatives of a knot is a topological knot invariant[2].
The invariant can also be computed using a grid diagram corresponding to a particular Legendrian representative of a knot[3][4]. In this setting, the number can be computed as the writhe of the diagram minus the number of 'northwest' corners.
By smoothing the 'northeast' and 'southwest' corners and rotating the diagram and switching all crossings, one can convert a grid diagram into the associated Legendrian knot.
References
- ^ "Entrelacements et équations de Pfaff". Astérisque. 107/108: 87–161. 1983. (Bennequin's doctoral dissertation)
- ^ Ng, Lenhard (2012). "On arc index and maximal thurston–bennequin number". Journal of Knot Theory and Its Ramifications. 21 (04): 1250031. arXiv:math/0612356. doi:10.1142/S0218216511009820. ISSN 0218-2165.
- ^ Ozsváth, Peter S.; Stipsicz, András I.; Szabó, Zoltán (2015). Grid Homology for Knots and Links. American Mathematical Society. pp. 220–221. ISBN 978-1-4704-3442-7.
- ^ Dynnikov, I.; Prasolov, M. (2013). "Bypasses for rectangular diagrams. A proof of the Jones conjecture and related questions". Transactions of the Moscow Mathematical Society. 74: 97–144. doi:10.1090/S0077-1554-2014-00210-7. ISSN 0077-1554.
- "Thurston–Bennequin number", The Knot Atlas.
- Lee Rudolph (1997). "The slice genus and the Thurston–Bennequin invariant of a knot". Proceedings of the American Mathematical Society. 125: 3049–3050. doi:10.1090/S0002-9939-97-04258-5. MR 1443854.