HOMFLY-PT homology
E665095
HOMFLY-PT homology is a triply graded link homology theory in knot theory whose graded Euler characteristic recovers the HOMFLY-PT polynomial, providing a powerful categorified invariant of links.
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
categorified knot invariant
ⓘ
link homology theory ⓘ |
| baseRing | typically defined over Q or C ⓘ |
| categorifies | HOMFLY-PT polynomial NERFINISHED ⓘ |
| constructedBy |
Lev Rozansky
NERFINISHED
ⓘ
Mikhail Khovanov NERFINISHED ⓘ |
| constructionMethod |
derived categories of coherent sheaves in some approaches
ⓘ
foam-based constructions ⓘ matrix factorizations ⓘ |
| definedFor | link diagrams via chain complexes ⓘ |
| field |
knot theory
ⓘ
low-dimensional topology ⓘ |
| functoriality | invariant under Reidemeister moves ⓘ |
| generalizes | Khovanov-Rozansky sl(N) homology NERFINISHED ⓘ |
| grading | triply graded ⓘ |
| gradingTypes |
a-grading
ⓘ
homological grading ⓘ q-grading ⓘ |
| hasApplication |
connections to algebraic geometry
ⓘ
connections to string theory and BPS state counting ⓘ distinguishing links with same HOMFLY-PT polynomial ⓘ relations to representation theory ⓘ study of knot concordance ⓘ |
| hasProperty |
depends only on isotopy class of link
ⓘ
functorial up to sign for link cobordisms in many constructions ⓘ graded Euler characteristic equals HOMFLY-PT polynomial ⓘ link invariant up to isomorphism of graded homology groups ⓘ refines HOMFLY-PT polynomial with more information ⓘ stronger than HOMFLY-PT polynomial as an invariant ⓘ triply graded over three integer gradings ⓘ |
| invariantOf |
links in R^3
ⓘ
oriented links in S^3 ⓘ |
| obtainedAs | homology of a triply graded chain complex ⓘ |
| relatedConcept |
colored HOMFLY-PT homology
ⓘ
superpolynomial of a knot ⓘ triply graded homology ⓘ |
| relatedTo |
Khovanov homology
NERFINISHED
ⓘ
categorification ⓘ quantum invariants of links ⓘ sl(N) link homology ⓘ |
| specializesTo |
Khovanov homology for N = 2
ⓘ
sl(N) link homology via spectral sequences ⓘ |
| usedIn |
categorical representation theory
ⓘ
refinement of quantum sl(N) invariants ⓘ topological quantum field theory constructions ⓘ |
| yearIntroduced | early 2000s ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.