Celebratio Mathematica — Fuchs — Legendrian Knot Theory

Dmitry Fuchs and Legendrian knot theory

Works connected to Yakov M. Eliashberg

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J. Epstein and D. Fuchs: “On the invariants of Legendrian mirror torus links,” pp. 103–115 in Symplectic and contact topology: Interactions and perspectives (Toronto and Montreal, 26 March–7 April 2001). Edited by Y. Eliashberg, B. Khesin, and F. Lalonde. Fields Institute Communications 35. American Mathematical Society (Providence, RI), 2003. MR 1969270 Zbl 1044.57007 incollection

We will compute several invariants of Legendrian mirror torus links in the standard contact space \( \{\mathbb{R}^3 \), \( y\,dx - dz\} \). Our purpose is not to classify Legendrian mirror torus links. A classification of Legendrian torus links, both algebraic and mirror, is contained in a recent paper by Etnyre and Honda [2001] who prove that topologically isotopic algebraic and mirror torus links with equal Thurston–Bennequin and Maslov numbers must be Legendrian isotopic. Nevertheless, the behavior of both classical and non-classical invariants of Legendrian mirror torus links is interesting, becuase it contradicts some natural conjectures and gives rise to new ones. In particular, we note that for \( q \leq p \), the behavior of the invariants of the Legendrian mirror torus link \( K(-p,q) \) depends on the parity of \( q \), where \( q \) is the braid index of \( K(-p,q) \).

François Lalonde	Related
Judith Karen Epstein	Related
Yakov M. Eliashberg	Related
Boris A. Khesin	Related