Spaces of Legendrian cables and Seifert fibered links

TL;DR

This work determines the homotopy types of spaces of Legendrian embeddings attaining maximal Thurston–Bennequin invariants, establishing a recursive description for max–tb Legendrian cables and a complete analysis for max–tb Legendrian Seifert fibered links in (S^3, ξstd). It proves that for sufficiently positive cables the space of contact structures on the knot complement is homotopy equivalent to that of the underlying knot, and it shows contractibility of contact-structure spaces for Legendrian Seifert fibrations over surfaces with boundary, yielding contractible contactomorphism groups up to the base mapping class group. The paper also demonstrates an injective h-principle for many families of Legendrian embeddings, including all max–tb Legendrian positive Seifert fibered links, via gluing-annuli techniques and π2-invariants for Legendrian loops (e.g., Kalman and Gramain type loops). Collectively, these results produce infinitely many new components in the spaces of Legendrian embeddings with maximal tb, reveal precise homotopy types (involving unitary factors and pure braid groups) for Seifert-related links, and connect the Legendrian theory with the topology of knot/link complements and their contact structures. The methods integrate convex surface theory, microfibration arguments, and detailed analysis of the complement geometry, advancing understanding of flexible/rigid phenomena in Legendrian topology.

Abstract

We determine the homotopy type of the spaces of several Legendrian knots and links with the maximal Thurston--Bennequin invariant. In particular, we give a recursive formula of the homotopy type of the space of Legendrian embeddings of sufficiently positive cables, and determine the homotopy type of the space of Legendrian embeddings of Seifert fibered links, which include all torus knots and links, in the standard contact 3-sphere, except when one of the link components is a negative torus knot. In general, we prove that the space of contact structures on the complement of a sufficiently positive Legendrian cable with the maximal Thurston-Bennequin invariant is homotopy equivalent to the space of contact structures on the complement of the underlying Legendrian knot, and prove that the space of contact structures on a Legendrian Seifert fibered space over a compact oriented surface with boundary is contractible. From this result, we find infinitely many new components of the space of Legendrian embeddings in the standard contact 3-sphere that satisfy an injective h-principle. These include the spaces of Legendrian embeddings of an algebraic link with the maximal Thurston--Bennequin invariant. In particular, the inclusion of these Legendrian embedding spaces into the corresponding formal Legendrian embedding spaces is a homotopy injection.

Figures (9)

• Figure 1: Bypass attachment
• Figure 2: Two attaching arcs of non-effective bypasses.
• Figure 3: An effective bypass attachment on an annulus
• Figure 4: Some possible holonomy of dividing curves on an annulus.
• Figure 5: Legendrian $(2,3)$-torus knot and $(4,6)$-torus link with the maximal Thurston--Bennequin number.

Theorems & Definitions (83)

• Remark 1.2
• Theorem 1.3
• Definition 1.4
• Corollary 1.5
• Example 1.6
• Theorem 1.7
• Corollary 1.8
• Theorem 1.9
• Remark 1.10
• Theorem 1.11