Alexander polynomial
E665082
The Alexander polynomial is a classical knot invariant in algebraic topology that assigns a Laurent polynomial to each knot or link, capturing essential information about its topological structure.
Observed surface forms (1)
| Surface form | Occurrences |
|---|---|
| unknot | 0 |
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
Laurent polynomial-valued invariant
ⓘ
knot invariant ⓘ link invariant ⓘ |
| appliesTo | links with multiple components ⓘ |
| canDistinguish | some non-equivalent knots ⓘ |
| cannotDistinguish | all non-equivalent knots ⓘ |
| captures |
topological information about a knot
ⓘ
topological information about a link ⓘ |
| category | classical knot invariant ⓘ |
| codomain | Z[t,t^{-1}] ⓘ |
| computableFrom |
Seifert surface of the knot
ⓘ
Wirtinger presentation of the knot group ⓘ |
| definedUsing |
Alexander module
NERFINISHED
ⓘ
Fox calculus NERFINISHED ⓘ Seifert matrix NERFINISHED ⓘ determinant of V - tV^T ⓘ first homology of the infinite cyclic cover ⓘ infinite cyclic cover of the knot complement ⓘ presentation matrix of the Alexander module ⓘ |
| degreeRelatedTo | twice the genus for fibered knots ⓘ |
| dependsOn | choice of knot or link ⓘ |
| evaluationProperty | Δ_K(1) = ±1 for a knot ⓘ |
| extension | multivariable Alexander polynomial for links ⓘ |
| field |
algebraic topology
ⓘ
knot theory ⓘ |
| generalizedBy |
higher-order Alexander invariants
ⓘ
twisted Alexander polynomial ⓘ |
| hasAlexanderPolynomial | 1 ⓘ |
| historicalPeriod | introduced in the 1920s ⓘ |
| input |
oriented knot
ⓘ
oriented link ⓘ |
| introducedBy | James Waddell Alexander II NERFINISHED ⓘ |
| invariantUnder |
Reidemeister moves
NERFINISHED
ⓘ
ambient isotopy of knots ⓘ |
| namedAfter | James Waddell Alexander II NERFINISHED ⓘ |
| normalizationProperty | defined up to multiplication by ±t^n ⓘ |
| output | Laurent polynomial in one variable ⓘ |
| relatedInvariant |
Conway polynomial
NERFINISHED
ⓘ
HOMFLY-PT polynomial NERFINISHED ⓘ Jones polynomial NERFINISHED ⓘ |
| specialCaseOf | multivariable Alexander polynomial NERFINISHED ⓘ |
| symmetryProperty | Δ_K(t) = ± t^n Δ_K(t^{-1}) ⓘ |
| usedToStudy |
3-manifold topology
ⓘ
fibered knots ⓘ knot concordance ⓘ knot genus ⓘ |
| valueFor | unknot ⓘ |
| variable | t ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.