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.
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 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.