Jones polynomial
E169187
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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Jones polynomial canonical | 5 |
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
Laurent polynomial–valued invariant
ⓘ
knot invariant ⓘ link invariant ⓘ |
| arisesFrom |
representations of braid groups
ⓘ
subfactor theory in von Neumann algebras ⓘ |
| canBeComputedFrom | braid representation of a link ⓘ |
| canBeComputedUsing |
Kauffman polynomial
ⓘ
surface form:
Kauffman bracket
|
| categorifiedBy | Khovanov homology ⓘ |
| codomain |
Laurent polynomials in a variable q^{1/2}
ⓘ
Laurent polynomials in a variable t^{1/2} ⓘ |
| connectedTo |
Chern–Simons theory
ⓘ
surface form:
Chern–Simons topological quantum field theory
Witten–Reshetikhin–Turaev invariant ⓘ
surface form:
Witten–Reshetikhin–Turaev invariants
quantum groups ⓘ |
| definedBy | skein relation at a crossing ⓘ |
| dependsOn | choice of orientation of link components ⓘ |
| distinguishes | many non-equivalent knots ⓘ |
| domain | oriented links in 3-dimensional space ⓘ |
| field |
knot theory
ⓘ
low-dimensional topology ⓘ |
| generalizationOf | Alexander polynomial in some contexts ⓘ |
| hasCoefficientRing | integers ⓘ |
| hasExponentType | half-integers in the variable exponent ⓘ |
| inspired |
categorification leading to Khovanov homology
ⓘ
development of quantum invariants of 3-manifolds ⓘ |
| introducedIn | paper on representations of braid groups ⓘ |
| invariantUnder |
Reidemeister moves
ⓘ
ambient isotopy of links ⓘ |
| namedAfter | Vaughan Jones ⓘ |
| normalizationCondition | value on the unknot equals 1 ⓘ |
| openProblem | whether the Jones polynomial detects the unknot ⓘ |
| property |
different knots can share the same Jones polynomial
ⓘ
not a complete knot invariant ⓘ |
| refines | classical knot invariants ⓘ |
| relatedConjecture | Volume conjecture ⓘ |
| relatedTo |
HOMFLY-PT polynomial
ⓘ
surface form:
HOMFLY polynomial
HOMFLY-PT polynomial ⓘ Kauffman polynomial ⓘ |
| satisfies |
behavior under connected sum of knots
ⓘ
multiplicativity under disjoint union up to normalization ⓘ skein relation ⓘ |
| usedIn |
distinguishing mirror-image knots in some cases
ⓘ
studying chirality of knots ⓘ topological quantum computation ⓘ |
| usedToDistinguish | trefoil knot from the unknot ⓘ |
| valueOnUnknot | 1 ⓘ |
| variableConvention |
q
ⓘ
t ⓘ |
| yearIntroduced | 1984 ⓘ |
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.