Computable Legendrian invariants

By Lenhard L. Ng. 2000

136 citations recorded by the source corpus.

Abstract We establish tools to facilitate the computation and application of the Chekanov–Eliashberg differential graded algebra (DGA), a Legendrian-isotopy invariant of Legendrian knots and links in standard contact three space. More specifically, we reformulate the DGA in terms of front projections, and introduce the characteristic algebra, a new invariant derived from the DGA. We use our techniques to distinguish between several previously indistinguishable Legendrian knots and links.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/39e468e4e12f40834e150b854ca05ab89a92fc0b

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Legendrian and Transversal Knots (2003, background): “n inequality is not sharp, as it only give an upper bound of 1. While no known bound on tbis always sharp the Kauﬀman polynomial FK frequently seems to be very good. For example Theorem 3.6 (Ng 2003, [67]). If K is a 2-bridge knot, then the Kauﬀman bound is sharp: tb(K) = dF K .Moreover, the Kauﬀman”
Framed knot contact homology (2004, unknown): “G T ] 2 2 M ay”
A combinatorial description of knot Floer homology (2006, background): “In a different direction, the relationship between our combinatorial description and Legendrian knots seems tantalizing: one wonders whether this is perhaps the hint of a connection with the holomorphic invariants of those objects, compare [2], [3], [15].”
Augmentations are sheaves (2015, methodology): “In order to prove ∂2 = 0, we will embed T in a simple (in the sense of [Ng03]) front diagram for some closed, oriented Legendrian link L so that (A(T ), ∂) is a sub-DGA of (A(L), ∂), and then observe that we already know that ∂2 = 0 in A(L).”
A Duality Exact Sequence for Legendrian Contact Homology (2008, background): “The exact sequence generalizes the duality for Legendrian knots in R3 [26] and leads to a refinement of the Arnold Conjecture for double points of an exact Lagrangian admitting a Legendrian lift with linearizable contact homology, first proved in [7].”
Generating families and Legendrian contact homology in the standard contact space (2008, background): “This was observed by Ng [14] who showed that xy- and xz-diagrams of the same knot, however different they may seem, may be made to look almost the same.”
Knot and braid invariants from contact homology II (2003, background): “d to know when two DGAs are stable tame isomorphic. This can be done through “computable” invariants of equivalence classes of DGAs, including Poincar´e polynomials [3] and the characteristic algebra [18]. In the case of braid and knot DGAs, it turns out that the ﬁrst order Poincar´e polynomial over”
Knot contact homology (2011, background): “This was introduced in Bourgeois, Ekholm and Eliashberg [6; 5]; see also Mishachev [23] and Ng [25] for precursors to this notion.”
Combinatorial knot contact homology and transverse knots (2010, background): “.6 may becompared to an analogous result in Legendrian contact homology, namely that the Chekanov–Eliashberg DGA for the Legendrian mirror of a Legendrian knot Λ is the opposite of the DGA for Λ. See [Ng03]. 5. Computationsand Applications In this section, we perform some calculations with transvers”
Legendrian knots, transverse knots and combinatorial Floer homology (2006, unknown): “53D12, 57R17, 57R58; 57M25”
Augmentations and rulings of Legendrian knots (2004, methodology): “-lattice. As shown in Figure 7(b), the lower strand must also exit the a-lattice without any further corners. Note that the dipped diagram comes from modifying the resolution of a “simple” front (see [14], Section 2.3) whose right cusps are all pushed out to the right, so any portion of a disk lying”
On the equivalence of Legendrian and transverse invariants in knot Floer homology (2011, background): “Among its other achievements, LCH provided the first examples of Legendrian non-simple knot types [Che02] and resolved the Legendrian mirror problem [Ng03].”
Cabling and transverse simplicity (2003, background): “Many other non-Legendrian-simple knot types have been found since then (see for example [ Ng , EH2]).”
Transverse knots distinguished by knot Floer homology (2007, background): “Subsequently Legendrian contact homology and other “nonclassical” Legendrian invariants have been used to find many other examples of knots which are not Legendrian simple [11, 23].”
Contact homology and one parameter families of Legendrian knots (2004, methodology): “Throughout the paper, we will use the term polynomial to mean non-commutative polynomial.”