Conway polynomial
E29419
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.
Observed surface forms (2)
| Surface form | Occurrences |
|---|---|
| Conway normalized Alexander polynomial | 1 |
| Conway–Alexander polynomial | 1 |
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
knot invariant
ⓘ
link invariant ⓘ polynomial invariant ⓘ |
| alsoKnownAs |
Conway polynomial
ⓘ
surface form:
Conway normalized Alexander polynomial
Conway polynomial ⓘ
surface form:
Conway–Alexander polynomial
|
| appliesTo | oriented links with any number of components ⓘ |
| canBeComputedBy |
Seifert matrix methods
ⓘ
skein relation ⓘ |
| captures |
information about knot chirality in some cases
ⓘ
information about knot linking for links ⓘ |
| codomain | Laurent polynomials in one variable ⓘ |
| coefficientInterpretation | lowest-degree nonzero coefficient relates to linking numbers for links ⓘ |
| coefficientProperty | coefficients are integers ⓘ |
| computationalComplexity | can be computed in polynomial time for fixed crossing number but is generally hard for large diagrams ⓘ |
| definedBy | Conway skein triple (L₊, L₋, L₀) ⓘ |
| definedOn | isotopy classes of oriented links in S³ ⓘ |
| degreeProperty | degree of ∇(K) is bounded above by twice the genus of K ⓘ |
| dependsOn | oriented link diagram ⓘ |
| doesNotCompletelyClassify | knots ⓘ |
| field |
knot theory
ⓘ
low-dimensional topology ⓘ |
| firstCoefficientProperty | constant term is 1 for knots ⓘ |
| firstNontrivialCoefficient | often encodes Arf invariant mod 2 for knots ⓘ |
| functorialityProperty | invariant under ambient isotopy ⓘ |
| generalizes | Alexander polynomial normalization ⓘ |
| inspired | later skein-theoretic definitions of other knot polynomials ⓘ |
| introducedIn | 1960s ⓘ |
| invariantOf |
oriented knots
ⓘ
oriented links ⓘ |
| isNot | complete knot invariant ⓘ |
| namedAfter |
John H. Conway
ⓘ
surface form:
John Horton Conway
|
| normalizationChoice | gives Alexander polynomial with symmetric normalization ⓘ |
| normalizationCondition | ∇(unknot) = 1 ⓘ |
| orientationProperty | independent of choice of orientation up to sign changes in variable ⓘ |
| relatedInvariant |
HOMFLY-PT polynomial
ⓘ
Jones polynomial ⓘ |
| relatedTo | Alexander polynomial ⓘ |
| relationToAlexanderPolynomial | Δ_K(t) = ∇_K(t^{1/2} − t^{−1/2}) up to normalization ⓘ |
| satisfies | skein relation ∇(L₊) − ∇(L₋) = z ∇(L₀) ⓘ |
| symmetryProperty | ∇(K)(z) = ∇(K)(−z) for many knots (evenness property related to Alexander polynomial) ⓘ |
| usedFor |
distinguishing non-equivalent knots
ⓘ
studying knot concordance ⓘ studying link splitting properties ⓘ |
| valueOnSplitUnion | for split union L₁ ⊔ L₂, ∇(L₁ ⊔ L₂) = 0 ⓘ |
| valueOnTrivialLink | for n-component trivial link with n>1, ∇ = 0 ⓘ |
| variable | z ⓘ |
| zeroCondition | vanishes for split links with more than one component ⓘ |
Referenced by (6)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Conway normalized Alexander polynomial
this entity surface form:
Conway–Alexander polynomial
subject surface form:
John H. Conway
subject surface form:
John H. Conway
subject surface form:
John Horton Conway