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.
All labels observed (3)
| Label | Occurrences |
|---|---|
| HOMFLY-PT homology canonical | 1 |
| Khovanov–Rozansky homology | 1 |
| sl(n) Khovanov–Rozansky homology | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T7450502 — 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.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: HOMFLY-PT homology Context triple: [HOMFLY-PT polynomial, categorifiedBy, HOMFLY-PT homology]
-
A.
Khovanov homology
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.
-
B.
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.
-
C.
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.
-
D.
Kauffman polynomial
The Kauffman polynomial is a two-variable knot invariant in knot theory that generalizes and extends the information captured by the Jones polynomial.
-
E.
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.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: HOMFLY-PT homology Target entity description: 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.
-
A.
Khovanov homology
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.
-
B.
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.
-
C.
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.
-
D.
Kauffman polynomial
The Kauffman polynomial is a two-variable knot invariant in knot theory that generalizes and extends the information captured by the Jones polynomial.
-
E.
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.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
categorified knot invariant
ⓘ
link homology theory ⓘ |
| baseRing | typically defined over Q or C ⓘ |
| categorifies | HOMFLY-PT polynomial ⓘ |
| constructedBy |
Lev Rozansky
ⓘ
Mikhail Khovanov ⓘ |
| 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
ⓘ
linked to:
Khovanov homology
|
| 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
ⓘ
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 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
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Input
Subject: HOMFLY-PT homology Description of subject: 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.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: HOMFLY-PT homology
linked to: HOMFLY-PT homology