Fox n-coloring of knots
E941102
Fox n-coloring of knots is a classical algebraic technique in knot theory that assigns colors (integers modulo n) to arcs of a knot diagram according to specific rules, producing an invariant useful for distinguishing non-equivalent knots.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Fox n-coloring of knots canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T11695097 — 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: Fox n-coloring of knots Context triple: [Ralph Fox, notableConcept, Fox n-coloring of knots]
-
A.
Kauffman polynomial
The Kauffman polynomial is a two-variable knot invariant in knot theory that generalizes and extends the information captured by the Jones polynomial.
-
B.
Wirtinger presentation of knot groups
The Wirtinger presentation of knot groups is a classical method in knot theory that describes the fundamental group of a knot complement using generators and relations derived from a knot diagram.
-
C.
Hoste–Thistlethwaite–Weeks knot tables
The Hoste–Thistlethwaite–Weeks knot tables are comprehensive, systematically generated lists of prime knots (and links) organized by crossing number, widely used as a modern extension and refinement of classical knot tabulations in knot theory.
-
D.
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.
-
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: Fox n-coloring of knots Target entity description: Fox n-coloring of knots is a classical algebraic technique in knot theory that assigns colors (integers modulo n) to arcs of a knot diagram according to specific rules, producing an invariant useful for distinguishing non-equivalent knots.
-
A.
Kauffman polynomial
The Kauffman polynomial is a two-variable knot invariant in knot theory that generalizes and extends the information captured by the Jones polynomial.
-
B.
Wirtinger presentation of knot groups
The Wirtinger presentation of knot groups is a classical method in knot theory that describes the fundamental group of a knot complement using generators and relations derived from a knot diagram.
-
C.
Hoste–Thistlethwaite–Weeks knot tables
The Hoste–Thistlethwaite–Weeks knot tables are comprehensive, systematically generated lists of prime knots (and links) organized by crossing number, widely used as a modern extension and refinement of classical knot tabulations in knot theory.
-
D.
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.
-
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 (39)
| Predicate | Object |
|---|---|
| instanceOf |
algebraic technique in knot theory
ⓘ
coloring invariant ⓘ knot invariant ⓘ |
| appliesTo |
knot diagrams
ⓘ
link diagrams ⓘ |
| canBeComputedBy | solving linear equations modulo n ⓘ |
| canBeComputedFrom | Wirtinger presentation of the knot group ⓘ |
| canBeFormulatedAs | system of linear equations over Z_n ⓘ |
| canDistinguish | non-equivalent knots ⓘ |
| dependsOn | choice of modulus n but not on particular diagram of the knot ⓘ |
| domain | oriented knots in S^3 ⓘ |
| field |
knot theory
ⓘ
low-dimensional topology ⓘ |
| generalizationOf | simple 3-coloring invariants of knots ⓘ |
| hasAlgebraicInterpretation | module over Z_n associated to the knot ⓘ |
| hasRule |
a coloring is nontrivial if not all arcs receive the same color
ⓘ
at each crossing, twice the color of the over-arc equals the sum of the colors of the under-arcs modulo n ⓘ each arc of the knot diagram is assigned a color in Z_n ⓘ |
| historicalContext | introduced in the mid-20th century ⓘ |
| invariantUnder | Reidemeister moves NERFINISHED ⓘ |
| isStableUnder | ambient isotopy of knots ⓘ |
| isTrivialIf | only monochromatic colorings exist ⓘ |
| namedAfter | Ralph H. Fox NERFINISHED ⓘ |
| nontrivialColoringImplies | knot determinant is divisible by n ⓘ |
| output |
cardinality of the set of valid colorings modulo n
ⓘ
set of all valid colorings modulo n ⓘ |
| relatedTo |
determinant of a knot
ⓘ
dihedral group representations of knot groups ⓘ first homology of the 2-fold branched cover of S^3 over the knot ⓘ homomorphisms from the knot group to the dihedral group D_{2n} ⓘ |
| requires | integer n ≥ 2 ⓘ |
| specialCase | 3-coloring of knots ⓘ |
| usedFor |
constructing simple examples in introductory knot theory
ⓘ
distinguishing the trefoil knot from the unknot ⓘ |
| uses |
arcs of a knot diagram
ⓘ
integers modulo n ⓘ knot diagrams ⓘ |
| yields |
knot invariant
ⓘ
number of distinct n-colorings of a knot diagram ⓘ |
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: Fox n-coloring of knots Description of subject: Fox n-coloring of knots is a classical algebraic technique in knot theory that assigns colors (integers modulo n) to arcs of a knot diagram according to specific rules, producing an invariant useful for distinguishing non-equivalent knots.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.