Bennequin type inequalities in Lens spaces (Q2883876)

The paper under review studies Bennequin-type inequalities for Legendrian and transverse knots and links in lens spaces. Just as in the classical setting, knot theory plays a prominent role in the study of contact 3-manifolds. A null-homologous Legendrian knot \(K\) comes equipped with a pair of classical invariants: the Thurston-Bennequin number and rotation number. Similarly, to a null-homologous transverse knot one can associate a classical invariant known as the self-linking number. It was shown in [\textit{K. L. Baker} and \textit{J. B. Etnyre}, Perspectives in analysis, geometry, and topology. On the occasion of the 60th birthday of Oleg Viro. Based on the Marcus Wallenberg symposium on perspectives in analysis, geometry, and topology, Stockholm, Sweden, May 19--25, 2008. Basel: Birkhäuser. Progress in Mathematics 296, 19--37 (2012; Zbl 1273.57006)] how to extend these invariants to the case where \(K\) is rationally null-homologous.NEWLINENEWLINEIf \((Y,\xi)\) is a tight contact 3-manifold and \(K\) is a null-homologous knot in \(Y\) with Seifert surface \(\Sigma\), \textit{Y. Eliashberg} [Topological methods in modern mathematics. Proceedings of a symposium in honor of John Milnor's sixtieth birthday, held at the State University of New York at Stony Brook, USA, June 14-June 21, 1991. Houston, TX: Publish or Perish, Inc. 171--193 (1993; Zbl 0809.53033)], generalizing work of \textit{D. Bennequin} [Astérisque 107/108, 87--161 (1983; Zbl 0573.58022)], proved the following bounds on the classical invariants for Legendrian or transverse representatives of \(K\): NEWLINE\[NEWLINE\mathrm{tb}(K_l) + |\mathrm{rot}(K)| \leq 2g(\Sigma) - 1 \;\;\;\; \text{sl}(K_t) \leq 2g(\Sigma) - 1.NEWLINE\]NEWLINENEWLINENEWLINESince that time, a multitude of bounds on the Thurston-Bennequin, rotation, and self-linking numbers have appeared for Legendrian knots in \((\mathbb{R}^3,\xi_{std})\). Ng showed in [\textit{L. Ng}, Int. Math. Res. Not. 2008, Article ID rnn116, 18 p. (2008; Zbl 1171.57007)] that many of these upper-bounds can be established via a single, unified approach. The article under review extends Ng's results to lens spaces equipped with their unique, universally tight contact structure.NEWLINENEWLINESpecifically, the author shows that if \(i\) is a \(\mathbb{Q}\)-valued invariant of oriented links in \(L(p,q)\) such that \(i(L_+) + 1 \leq \max(i(L_-) - 1, i(L_0))\), and \(i(L_-) - 1 \leq \max(i(L_+) + 1, i(L_0))\), where \((K_+,K_-,K_0)\) denotes the usual Skein triple which differs only at a single crossing, and if \(\mathrm{sl}_{\mathbb{Q}}(T(\tau)) \leq -i(\tau)\) for a certain class of ``trivial links'', then the maximal self linking number of a knot (or link) type \(K\) is bounded above by \(-i(K)\).NEWLINENEWLINEOne corollary of the above result is an extension of the Franks-Williams-Morton inequality, which provides a bound the maximal self-linking number coming from the HOMFLY-PT polynomial. Another corollary is a well-known result of \textit{R. Fintushel} and \textit{R. Stern} [Math. Z. 178, 143 (1981; Zbl 0466.57005)] establishing that if surgery on a knot in \(L(p,q)\) yields \(S^3\), then \(\pm q\) is a quadratic residue mod \(p\).NEWLINENEWLINENEWLINENEWLINE The proof of the author's main result goes by way of grid diagrams, defined for lens space knots and links in [\textit{K. L. Baker} and \textit{J. E. Grigsby}, J. Symplectic Geom. 7, No. 4, 415--448 (2009; Zbl 1205.57007)]. It relies on earlier work of the author [J. Knot Theory Ramifications 21, No. 6, 1250060, 31 p. (2012; Zbl 1251.57014)] and makes extensive use of the combinatorial structure inherent to grid diagrams.