Knot and braid invariants from contact homology II

By Lenhard L. Ng. 2003

76 citations recorded by the source corpus.

We present a topological interpretation of knot and braid contact homology in degree zero, in terms of cords and skein relations. This interpretation allows us to extend the knot invariant to embedded graphs and higher-dimensional knots. We calculate the knot invariant for two-bridge knots and relate it to double branched covers for general knots. In the appendix we show that the cord ring is determined by the fundamental group and peripheral structure of a knot and give applications.

Subfields: Geometry and Topology, Mathematical Physics, Neurology

View Paper: https://www.semanticscholar.org/paper/6907636186732a0ed1f7e550f3ab4eeb5eed0e13

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Citations in the corpus

A topological introduction to knot contact homology (2012, methodology): “Definition 3.9 ([Ng05a]).”
Knot contact homology (2011, background): “1 confirms the conjecture that the combinatorial knot invariant defined and studied by the third author in [26; 27; 29], called “knot contact homology” in those works, indeed equals the Legendrian homology of the conormal lift.”
Combinatorial knot contact homology and transverse knots (2010, methodology): “This is clear for the first term in braces; by the expression for ΦLσk from [Ng05a, Lemma 4.6], the second term in braces is the 0 matrix except in the kk entry, where it is ak+1,k − φB(ak+1,k) (or λµ −w(B)a21 − φB(a21) if k = 1), which is a scalar multiple of the k + 1, k entry of the matrix A −ΛB…”
Legendrian contact homology in $P \times \mathbb{R}$ (2007, methodology): “This construction is the inspiration behind Ng’s exciting new invariants of knots in R(3), [11, 12, 13].”
Morse flow trees and Legendrian contact homology in 1-jet spaces (2005, unknown): “ar X iv :m at h/ 05 09 38 6v 1 [ m at h. SG ] 1 6 Se p 20”
Legendrian Contact Homology in P X R (2005, methodology): “When n = 3, this application is the first step in extending and providing a contact geometric underpinning for the new knot invariants of Ng [11, 12, 13].”
Legendrian and Transversal Knots (2003, methodology): “Here we describe the some of the geometric idea behind the braid invariants and refer the reader to [65, 66] for details on knot invariant and the combinatorial proofs of invariance.”