Instanton Floer homology for knots via 3-orbifolds

Collin, Olivier; Steer, B.

doi:10.4310/jdg/1214425027

Cited by 27 publications

(37 citation statements)

References 37 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The formula (2) then follows by combining this result with the 2-functor axiom for 1-morphisms in (3).…”

Section: -Morphisms In Floermentioning

confidence: 98%

“…For example, in [22] we investigate the topological quantum field theory with corners (roughly speaking; not all the axioms are satisfied) in 2 + 1 + 1 dimensions arising from moduli spaces of flat bundles with compact structure group on punctured surfaces and three-dimensional cobordisms containing tangles. In particular, this gives rise to SU (N ) Floer theoretic invariants for 3-manifolds that should be thought of as Lagrangian Floer versions of gauge-theoretic invariants investigated by Donaldson and Floer, in the case without knots, and Kronheimer-Mrowka [9] and Collin-Steer [3], in the case with knots. The construction of such theories was suggested by Fukaya in [4] and was one of the motivations for the development of Fukaya categories.…”

Section: -Morphisms In Floermentioning

confidence: 99%

“…The composition of 1-morphisms in this category is concatenation, which we denote by #. The construction of the functor Φ(L 01 ) above extends to a categorification 2-functor to the 2-category of categories (3) Floer # −→ Cat .…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Functoriality for Lagrangian correspondences in Floer theory

Wehrheim¹,

Woodward²

2010

Quantum Topol.

141

View full text Add to dashboard Cite

Abstract. We associate to every monotone Lagrangian correspondence a functor between Donaldson-Fukaya categories. The composition of such functors agrees with the functor associated to the geometric composition of the correspondences, if the latter is embedded. That is "categorification commutes with composition" for Lagrangian correspondences. This construction fits into a symplectic 2-category with a categorification 2-functor, in which all correspondences are composable, and embedded geometric composition is isomorphic to the actual composition. As a consequence, any functor from a bordism category to the symplectic category gives rise to a category valued topological field theory.

show abstract

“…The formula (2) then follows by combining this result with the 2-functor axiom for 1-morphisms in (3).…”

Section: -Morphisms In Floermentioning

confidence: 98%

Section: -Morphisms In Floermentioning

confidence: 99%

See 1 more Smart Citation

Functoriality for Lagrangian correspondences in Floer theory

Wehrheim¹,

Woodward²

2010

Quantum Topol.

141

View full text Add to dashboard Cite

show abstract

“…This could be further pursued from several angles. For example, this result could be viewed as motivation for studying knot invariants in gauge-theoretic contexts, such as Seiberg-Witten or Donaldson's theories (compare Collin-Steer [2]). In a different direction, there seems to be a close connection between knot Floer homology and Khovanov's homology for links (cf Khovanov [4], Khovanov-Rozansky [5], Bar-Natan [1] and Lee [7]), see also recent work of Rasmussen [22].…”

Section: Introductionmentioning

confidence: 99%

Knot Floer homology and integer surgeries

Ozsváth

Szabó

2008

Algebr. Geom. Topol.

120

185

View full text Add to dashboard Cite

Let Y be a closed three-manifold with trivial first homology, and let K Y be a knot. We give a description of the Heegaard Floer homology of integer surgeries on Y along K in terms of the filtered homotopy type of the knot invariant for K . As an illustration of these techniques, we calculate the Heegaard Floer homology groups of non-trivial circle bundles over Riemann surfaces (with coefficients in Z=2Z).

show abstract

“…This is precisely the definition of the knot invariant which was introduced and studied in [39,40,41] (see also [42,43]). This invariant, sometimes called Casson-Lin invariant, is well-defined away from the roots of the Alexander polynomial of K and turns out to be equal to the linear combination of more familiar invariants, α ∈ [0, 1], λ α (Y ; K) was constructed in [44] (see also [45,46]) and, therefore, is expected to be the same as (3.8).…”

Section: Surface Operatorsmentioning

confidence: 99%

Gauge theory and knot homologies

Gukov

2007

Fortschritte der Physik

104

View full text Add to dashboard Cite

Topological gauge theories in four dimensions which admit surface operators provide a natural framework for realizing homological knot invariants. Every such theory leads to an action of the braid group on branes on the corresponding moduli space. This action plays a key role in the construction of homological knot invariants. We illustrate the general construction with examples based on surface operators in N = 2 and N = 4 twisted gauge theories which lead to a categorification of the Alexander polynomial, the equivariant knot signature, and certain analogs of the Casson invariant.

show abstract

Instanton Floer homology for knots via 3-orbifolds

Cited by 27 publications

References 37 publications

Functoriality for Lagrangian correspondences in Floer theory

Functoriality for Lagrangian correspondences in Floer theory

Knot Floer homology and integer surgeries

Gauge theory and knot homologies

Contact Info

Product

Resources

About