HOMFLY-PT polynomial
E171994
The HOMFLY-PT polynomial is a powerful knot and link invariant in knot theory that generalizes both the Alexander and Jones polynomials.
All labels observed (4)
| Label | Occurrences |
|---|---|
| HOMFLY-PT polynomial canonical | 18 |
| HOMFLY polynomial | 3 |
| HOMFLY-PT polynomial and its generalizations | 1 |
| HOMFLYPT polynomial | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T1483881 — 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 polynomial Context triple: [Conway polynomial, relatedInvariant, HOMFLY-PT polynomial]
-
A.
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.
-
B.
Conway polynomial
The Conway polynomial is an invariant of knots and links in topology that assigns a polynomial to each knot, capturing essential information about its structure and helping distinguish non-equivalent knots.
-
C.
Dowker–Thistlethwaite notation
Dowker–Thistlethwaite notation is a numerical encoding system used in knot theory to uniquely represent knot diagrams and facilitate their classification and study.
-
D.
Conway skein triple (L₊, L₋, L₀)
The Conway skein triple (L₊, L₋, L₀) is a standard configuration of three related link diagrams used in knot theory to express how a link invariant, such as the Conway polynomial, changes under local crossing modifications.
-
E.
Conway notation for knots
Conway notation for knots is a mathematical system introduced by John H. Conway that encodes knot and link diagrams into concise symbolic expressions to classify and study them.
- 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 polynomial Target entity description: The HOMFLY-PT polynomial is a powerful knot and link invariant in knot theory that generalizes both the Alexander and Jones polynomials.
-
A.
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.
-
B.
Conway polynomial
The Conway polynomial is an invariant of knots and links in topology that assigns a polynomial to each knot, capturing essential information about its structure and helping distinguish non-equivalent knots.
-
C.
Dowker–Thistlethwaite notation
Dowker–Thistlethwaite notation is a numerical encoding system used in knot theory to uniquely represent knot diagrams and facilitate their classification and study.
-
D.
Conway skein triple (L₊, L₋, L₀)
The Conway skein triple (L₊, L₋, L₀) is a standard configuration of three related link diagrams used in knot theory to express how a link invariant, such as the Conway polynomial, changes under local crossing modifications.
-
E.
Conway notation for knots
Conway notation for knots is a mathematical system introduced by John H. Conway that encodes knot and link diagrams into concise symbolic expressions to classify and study them.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
knot invariant
ⓘ
link invariant ⓘ polynomial invariant ⓘ |
| alsoKnownAs |
HOMFLY polynomial
ⓘ
linked to:
HOMFLY-PT polynomial
HOMFLYPT polynomial ⓘ
linked to:
HOMFLY-PT polynomial
|
| categorifiedBy | HOMFLY-PT homology ⓘ |
| codomain | Laurent polynomials in two variables ⓘ |
| definedFor | oriented links ⓘ |
| dependsOn | two variables ⓘ |
| distinguishes | many non-equivalent knots ⓘ |
| domain | isotopy classes of oriented links in S^3 ⓘ |
| field |
knot theory
ⓘ
low-dimensional topology ⓘ |
| generalizes |
Alexander polynomial
ⓘ
Jones polynomial ⓘ |
| hasProperty |
multiplicative under disjoint union of links
ⓘ
sensitive to orientation of components ⓘ |
| introducedBy |
Andrew Ocneanu
ⓘ
David Yetter ⓘ Jim Hoste ⓘ Kenneth Millett ⓘ Peter Freyd ⓘ W. B. R. Lickorish ⓘ |
| introducedIndependentlyBy |
Józef H. Przytycki
ⓘ
Paweł Traczyk ⓘ |
| invariantUnder |
Reidemeister moves
ⓘ
ambient isotopy ⓘ |
| namedAfter |
Freyd
ⓘ
Hoste ⓘ Lickorish ⓘ Milnor ⓘ Ocneanu ⓘ Przytycki ⓘ Traczyk ⓘ Yetter ⓘ |
| normalizationCondition | value 1 on the unknot ⓘ |
| reducesTo |
Alexander polynomial under specialization
ⓘ
Jones polynomial under specialization ⓘ |
| relatedTo |
Chern–Simons theory
ⓘ
quantum groups ⓘ |
| satisfies | skein relation ⓘ |
| usedToStudy |
knot concordance
ⓘ
link cobordism ⓘ |
| variableNotation |
a
ⓘ
l ⓘ m ⓘ q ⓘ |
| yearIntroduced | 1985 ⓘ |
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
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: HOMFLY-PT polynomial Description of subject: The HOMFLY-PT polynomial is a powerful knot and link invariant in knot theory that generalizes both the Alexander and Jones polynomials.
Referenced by (23)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: HOMFLY-PT polynomial
linked to: HOMFLY-PT polynomial
linked to: HOMFLY-PT polynomial
linked to: HOMFLY-PT polynomial
subject linked to:
Przytycki
linked to: HOMFLY-PT polynomial