HOMFLY-PT polynomial

E171994

knot invariant link invariant polynomial invariant

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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All labels observed (3)

Label	Occurrences
HOMFLY polynomial	3
HOMFLY-PT polynomial canonical	3
HOMFLYPT polynomial	1

Predicate	Object
instanceOf	knot invariant ⓘ link invariant ⓘ polynomial invariant ⓘ
alsoKnownAs	HOMFLY-PT polynomial ⓘ surface form: HOMFLY polynomial HOMFLY-PT polynomial ⓘ surface form: HOMFLYPT 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 ⓘ

Full triples — surface form annotated when it differs from this entity's canonical label.

Conway polynomial → relatedInvariant → HOMFLY-PT polynomial ⓘ

Jones polynomial → relatedTo → HOMFLY-PT polynomial ⓘ

this entity surface form: HOMFLY polynomial

Conway skein triple (L₊, L₋, L₀) → relatedTo → HOMFLY-PT polynomial ⓘ

HOMFLY-PT polynomial → alsoKnownAs → HOMFLY-PT polynomial ⓘ

this entity surface form: HOMFLY polynomial

HOMFLY-PT polynomial → alsoKnownAs → HOMFLY-PT polynomial ⓘ

this entity surface form: HOMFLYPT polynomial

Chern–Simons theory → relatedTo → HOMFLY-PT polynomial ⓘ

this entity surface form: HOMFLY polynomial