Khovanov homology
E656685
Khovanov homology is a powerful link invariant in knot theory that lifts the Jones polynomial to a graded homology theory, providing stronger topological information than the polynomial alone.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Khovanov homology canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T7338471 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
Target entity: Khovanov homology Context triple: [Jones polynomial, categorifiedBy, Khovanov homology]
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A.
Witten–Reshetikhin–Turaev invariant
The Witten–Reshetikhin–Turaev invariant is a quantum invariant of 3-manifolds and links derived from Chern–Simons theory and quantum groups, playing a central role in low-dimensional topology and quantum topology.
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B.
Jones polynomial
The Jones polynomial is a powerful knot invariant in topology that assigns to each knot or link a Laurent polynomial, enabling the distinction of many knots that are indistinguishable by classical invariants.
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C.
Donaldson invariants
Donaldson invariants are sophisticated topological invariants of smooth four-dimensional manifolds derived from moduli spaces of anti-self-dual connections, central to the study of 4-manifold differential topology.
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D.
HOMFLY-PT polynomial
The HOMFLY-PT polynomial is a powerful knot and link invariant in knot theory that generalizes both the Alexander and Jones polynomials.
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E.
Kazhdan–Lusztig theory
Kazhdan–Lusztig theory is a framework in representation theory and algebraic geometry that studies Hecke algebras and their bases via Kazhdan–Lusztig polynomials, with deep connections to the representation theory of Lie algebras and geometry of Schubert varieties.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
Target entity: Khovanov homology Target entity description: Khovanov homology is a powerful link invariant in knot theory that lifts the Jones polynomial to a graded homology theory, providing stronger topological information than the polynomial alone.
-
A.
Witten–Reshetikhin–Turaev invariant
The Witten–Reshetikhin–Turaev invariant is a quantum invariant of 3-manifolds and links derived from Chern–Simons theory and quantum groups, playing a central role in low-dimensional topology and quantum topology.
-
B.
Jones polynomial
The Jones polynomial is a powerful knot invariant in topology that assigns to each knot or link a Laurent polynomial, enabling the distinction of many knots that are indistinguishable by classical invariants.
-
C.
Donaldson invariants
Donaldson invariants are sophisticated topological invariants of smooth four-dimensional manifolds derived from moduli spaces of anti-self-dual connections, central to the study of 4-manifold differential topology.
-
D.
HOMFLY-PT polynomial
The HOMFLY-PT polynomial is a powerful knot and link invariant in knot theory that generalizes both the Alexander and Jones polynomials.
-
E.
Kazhdan–Lusztig theory
Kazhdan–Lusztig theory is a framework in representation theory and algebraic geometry that studies Hecke algebras and their bases via Kazhdan–Lusztig polynomials, with deep connections to the representation theory of Lie algebras and geometry of Schubert varieties.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
categorification
ⓘ
homology theory ⓘ knot invariant ⓘ link invariant ⓘ |
| basedOn | Jones polynomial NERFINISHED ⓘ |
| categorifies | Jones polynomial NERFINISHED ⓘ |
| coefficientRing |
finite fields
ⓘ
integers ⓘ rational numbers ⓘ |
| definedFrom | enhanced states of a link diagram ⓘ |
| detects | more topological information than the Jones polynomial ⓘ |
| domain |
link diagrams
ⓘ
oriented links in S^3 ⓘ |
| field |
algebraic topology
ⓘ
knot theory ⓘ low-dimensional topology ⓘ representation theory ⓘ |
| functorialWithRespectTo | link cobordisms ⓘ |
| generalizedBy | Khovanov–Rozansky homology NERFINISHED ⓘ |
| hasApplication |
construction of Rasmussen invariant
ⓘ
study of knot concordance ⓘ study of slice genus ⓘ |
| hasGrading |
homological grading
ⓘ
q-grading ⓘ |
| hasObjectType |
bigraded abelian groups
ⓘ
bigraded vector spaces ⓘ |
| hasProperty |
bigraded
ⓘ
functorial ⓘ graded ⓘ invariant under Reidemeister moves ⓘ link invariant up to isomorphism ⓘ |
| hasVariant |
colored Khovanov homology
NERFINISHED
ⓘ
odd Khovanov homology NERFINISHED ⓘ reduced Khovanov homology ⓘ sl(n) Khovanov–Rozansky homology NERFINISHED ⓘ |
| inspired | developments in categorification ⓘ |
| introducedBy | Mikhail Khovanov NERFINISHED ⓘ |
| isStrongerInvariantThan | Jones polynomial NERFINISHED ⓘ |
| publishedIn | Journal of Differential Geometry NERFINISHED ⓘ |
| relatedTo |
Heegaard Floer homology
NERFINISHED
ⓘ
categorified quantum sl(2) NERFINISHED ⓘ knot Floer homology ⓘ quantum groups ⓘ |
| usedToDefine | s-invariant of a knot ⓘ |
| usesConstruction |
Frobenius algebra
NERFINISHED
ⓘ
chain complex ⓘ cube of resolutions ⓘ |
| yearOfIntroduction | 1999 ⓘ |
| yields | Jones polynomial as graded Euler characteristic ⓘ |
How these facts were elicited
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Subject: Khovanov homology Description of subject: Khovanov homology is a powerful link invariant in knot theory that lifts the Jones polynomial to a graded homology theory, providing stronger topological information than the polynomial alone.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.