Transverse string topology and the cord algebra

Basu, Somnath; McGibbon, Jason Frederick; Sullivan, Dennis; Sullivan, Michael C.

doi:10.4310/jsg.2015.v13.n1.a1

Cited by 2 publications

(15 citation statements)

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“…Remark 2.11. As already mentioned in the introduction, in [2] Basu, McGibbon, Sullivan and Sullivan have given a string topology description of a version of the cord algebra for a codimension 2 submanifold K ⊂ Q of some ambient manifold Q, proving a theorem which formally looks quite similar to Proposition 2.9. In the language we use here, the main difference in their work is the absence of N -strings, so that for knots K ⊂ R 3 the version of H string 0 (K) they define only recovers the specialization at λ = 1 of (the commutative version of) Cord(K).…”

Section: 2mentioning

confidence: 91%

“…We prove part (i). For a generic knot K parametrized by γ : S 1 = R/LZ → R 3 , the first four derivatives (γ, γ (2) , γ (3) , γ (4) ) span R 3 at each t ∈ S 1 . (For this, use the jet transversality theorem [25, Chapter 3] to make the corresponding map S 1 → (R 3 ) 4 transverse to the codimension two subset consisting of quadruples of vectors that lie in a plane.)…”

Section: 2mentioning

confidence: 99%

“…Finally, we consider similar maps when the target space is two dimensional. into the compactified moduli space of disks of type [2,0]. This map is an immersion of a 1-dimensional manifold into a 2-manifold with boundary with corners.…”

Section: Compactification Of Moduli Spaces and Gluingmentioning

confidence: 99%

“…Let M s+1 denote the image of this immersion. Then M s+1 consists of those disks for which some point in the (s + 1) th boundary arc hits K. Then M s+1 and M t+1 intersect (self-intersect if s = t) transversely at disks with two points hitting K (this corresponds to disks of type [2,2]). The boundary of M s consists of points in the codimension one boundary of M corresponding to disks as in Theorem 10.4 (a) and (b) as well as to interior points corresponding to disks of type [2, 3 2 ] as in Theorem 10.4 (c).…”

Section: Compactification Of Moduli Spaces and Gluingmentioning

confidence: 99%

“…Motivated by the general picture sketched by the 1 However, we note that it is an interesting open problem to find a similar topological interpretation of the full degree 0 knot contact homology as a Z[λ ±1 , µ ±1 , U ±1 ]-algebra. 2 In the presence of contractible closed geodesics in Q, this will require augmentations by holomorphic planes in T * Q, see e.g. [6].…”

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Knot contact homology, string topology, and the cord algebra

Cieliebak¹,

Ekholm²,

Latschev³

et al. 2017

Journal De L’École Polytechnique — Mathématiques

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The conormal Lagrangian L K of a knot K in R 3 is the submanifold of the cotangent bundle T * R 3 consisting of covectors along K that annihilate tangent vectors to K. By intersecting with the unit cotangent bundle S * R 3 , one obtains the unit conormal Λ K , and the Legendrian contact homology of Λ K is a knot invariant of K, known as knot contact homology. We define a version of string topology for strings in R 3 ∪ L K and prove that this is isomorphic in degree 0 to knot contact homology. The string topology perspective gives a topological derivation of the cord algebra (also isomorphic to degree 0 knot contact homology) and relates it to the knot group. Together with the isomorphism this gives a new proof that knot contact homology detects the unknot. Our techniques involve a detailed analysis of certain moduli spaces of holomorphic disks in T * R 3 with boundary on R 3 ∪ L K . 1 arXiv:1601.02167v2 [math.SG] 23 May 2017 * (K).In its most general form (see [11,33]), knot contact homology is the homology of a differential graded algebra over the group ringwhere the images of λ, µ under the connecting homomorphism generate H 1 (Λ K ) = H 1 (T 2 ) and U generates H 2 (S * R 3 ). The isomorphism class of H contact * (K) as a Z[λ ±1 , µ ±1 , U ±1 ]-algebra is then an isotopy invariant of the framed oriented knot K.The topological content of knot contact homology has been much studied in recent years; see for instance [1] for a conjectured relation, which we will not discuss here, to colored HOMFLY-PT polynomials and topological strings. One part of knot contact homology that has an established topological interpretation is its U = 1 specialization. In [31,32], the third author constructed a knot invariant called the cord algebra Cord(K), whose definition we will review in Section 2.2. The combined results of [31,32,15] then prove that the cord algebra is isomorphic as a Z[λ ±1 , µ ±1 ]-algebra to the U = 1 specialization of degree 0 knot contact homology. We will assume throughout this paper that we have set U = 1; 1 then the result is: 1 ([31, 32, 15]). H contact 0 (K) ∼ = Cord(K).It has been noticed by many people that the definition of the cord algebra bears a striking resemblance to certain operations in string topology [4,36]. Indeed, Basu, McGibbon, Sullivan, and Sullivan used this observation in [2] to construct a theory called "transverse string topology" associated to any codimension 2 knot K ⊂ Q, and proved that it determines the U = λ = 1 specialization of the cord algebra.In this paper, we present a different approach to knot contact homology and the cord algebra via string topology. Motivated by the general picture sketched by the 1 However, we note that it is an interesting open problem to find a similar topological interpretation of the full degree 0 knot contact homology as a Z[λ ±1 , µ ±1 , U ±1 ]-algebra. 2 In the presence of contractible closed geodesics in Q, this will require augmentations by holomorphic planes in T * Q, see e.g. [6].

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Section: 2mentioning

confidence: 91%

Section: 2mentioning

confidence: 99%

Section: Compactification Of Moduli Spaces and Gluingmentioning

confidence: 99%

Section: Compactification Of Moduli Spaces and Gluingmentioning

confidence: 99%

mentioning

confidence: 99%

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