Computing the Thurston–Bennequin invariant in open books

Abstract

Abstract. We give explicit formulas and algorithms for the computation of the Thurston-Bennequin invariant of a nullhomologous Legendrian knot on a page of a contact open book and on Heegaard surfaces in convex position. Furthermore, we extend the results to rationally nullhomologous knots in arbitrary 3-manifolds.

The Legendrian knot complement problem

“…Observe that this result does not depend on the choice of a. , for the fact that this is in general well defined see [12,Section 5]). With the same notation as from the first sections, one gets the following generalizations of these results.…”

“…Building up on this and using the formulas in [34,24,5] one can also get formulas for computing rotation numbers of Legendrian knots, self-linking numbers of transverse knots and the d 3 -invariant of the resulting contact manifold in general contact (1/n)-surgery diagrams along Legendrian knots [10]. In [12,11] similar formulas are given for computing the classical invariants of Legendrian knots sitting on the page of a contact open book.…”

“…A knot L 0 in M is called rationally nullhomologous if there exists a natural number k ∈ N such that k[L 0 ] = 0 ∈ H 1 (M ; Z). For rationally nullhomologous Legendrian knots in contact 3-manifolds one can generalize the Thurston-Bennequin invariant to the so-called rational Thurston-Bennequin invariant tb Q (see for example [2, Definition 6.2] or [24, Section 2 and 3], for the fact that this is in general well defined see [12,Section 5]).…”

Computing rotation and self-linking numbers in contact surgery diagrams

“…It is obvious that every rationally nullhomologous knot has a Seifert framing. Moreover, the Seifert framing is unique (see [7]).…”

Rotation numbers and the Euler class in open books

“…The rotation and Thurston-Bennequin numbers are defined for oriented nullhomologous Legendrian knots in arbitrary contact 3-manifolds. In the case of a Legendrian knot Λ in the standard contact 3-space, which we denote by (R 3 , ξ std ), these can be computed via simple combinatorial formulas from either the front or Lagrangian projection of Λ. Analogous techniques have been developed for Legendrians in special classes of 3-manifolds that admit natural notions of projections, and or in the case of Legendrians embedded in a single page of an open book [On18,EO08,DKK16,DK18]. This paper establishes techniques for computing the classical invariants of a generic Legendrian knot in an arbitrary contact 3-manifold.…”