An atlas of Legendrian knots

Abstract

We present an atlas of Legendrian knots in standard contact three-space. This gives a conjectural Legendrian classification for all knots with arc index at most 9, including alternating knots through 7 crossings and nonalternating knots through 9 crossings. Our method involves a computer search of grid diagrams and applies to transverse knots as well. The atlas incorporates a number of new, small examples of phenomena such as transverse nonsimplicity and non-maximal non-destabilizable Legendrian knots, and gives rise to new infinite families of transversely nonsimple knots.

44 pages, including 25 pages of tables

References in corpus (10)

• Legendrian knots, transverse knots and combinatorial Floer homology
• Combinatorial knot contact homology and transverse knots
• On arc index and maximal Thurston-Bennequin number
• Heegaard Floer invariants of Legendrian knots in contact three--manifolds
• Legendrian and transverse cables of positive torus knots
• Grid Diagrams, Braids, and Contact Geometry
• Rational Symplectic Field Theory for Legendrian knots
• A family of transversely nonsimple knots
• Legendrian contact homology and nondestabilizability
• Product Structures for Legendrian Contact Homology

Cited by in corpus (24)

• Legendrian knots and constructible sheaves
• Combinatorial knot contact homology and transverse knots
• Legendrian and transverse cables of positive torus knots
• Augmentations are Sheaves
• Obstructions to Lagrangian Cobordisms between Legendrians via Generating Families
• The contact homology of Legendrian knots with maximal Thurston-Bennequin invariant
• Monopole Floer homology and Legendrian knots
• A topological introduction to knot contact homology
• The augmentation category map induced by exact Lagrangian cobordisms
• The Surgery Unknotting Number of Legendrian Links
• Rectangular diagrams of surfaces: representability
• Distinguishing Legendrian knots with trivial orientation-preserving symmetry group
• Rectangular diagrams of surfaces: distinguishing Legendrian knots
• Spaces of tight contact structures in dimension 3
• GridPyM: a Python module to handle grid diagrams
• Classification of Legendrian knots of topological type $7_6$ with maximal Thurston--Bennequin number
• On the Lagrangian fillability of almost positive links
• Non-Orientable Lagrangian Cobordisms between Legendrian Knots
• Legendrian Rack Invariants of Legendrian Knots
• Transverse-Legendrian links
• Grid diagram for singular links
• On non-geometric augmentations in high dimensions
• Obstruction Complexes in Grid Homology
• Bounds on the mosaic number of Legendrian Knots