Fox n-coloring of knots

E941102

algebraic technique in knot theory coloring invariant knot invariant

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.

Try in SPARQL Jump to: Statements Referenced by

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 ⓘ

Referenced by (1)

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

Ralph Fox → notableConcept → Fox n-coloring of knots ⓘ