Fox calculus
E941103
Fox calculus is an algebraic tool in combinatorial group theory that uses formal derivatives to study group presentations and their topological applications.
Statements (45)
| Predicate | Object |
|---|---|
| instanceOf |
algebraic tool
ⓘ
mathematical theory ⓘ method in combinatorial group theory ⓘ |
| alsoKnownAs | free differential calculus ⓘ |
| appearsIn | Ralph H. Fox’s papers on free differential calculus ⓘ |
| basedOn |
free groups
ⓘ
group rings ⓘ |
| defines | Fox derivative NERFINISHED ⓘ |
| domain |
algebraic topology
ⓘ
geometric group theory ⓘ |
| field | combinatorial group theory ⓘ |
| formalism | noncommutative differential calculus on group rings ⓘ |
| generalizes | classical derivative rules to group rings ⓘ |
| hasApplicationIn |
3-manifold topology
ⓘ
knot theory ⓘ topology ⓘ |
| hasRule |
derivative of inverses in group rings
ⓘ
product rule for Fox derivatives ⓘ |
| influenced | later developments in noncommutative calculus on groups ⓘ |
| introducedBy | Ralph H. Fox NERFINISHED ⓘ |
| keyConcept |
Fox free derivative
ⓘ
augmentation map ⓘ derivation rules on group rings ⓘ |
| mathematicsSubjectClassification |
20F05
ⓘ
57M25 ⓘ |
| operatesOn |
free group on generators
ⓘ
integral group ring of a free group ⓘ |
| relatedTo |
Magnus expansion
NERFINISHED
ⓘ
Reidemeister torsion NERFINISHED ⓘ presentation matrices of modules ⓘ |
| studies | group presentations ⓘ |
| usedBy |
group theorists
ⓘ
topologists ⓘ |
| usedFor |
computing Alexander invariants
ⓘ
computing Alexander polynomials ⓘ computing Jacobian-like matrices for group presentations ⓘ computing relations among generators in a group presentation ⓘ deriving algebraic invariants from CW-complexes ⓘ studying chain complexes ⓘ studying covering spaces ⓘ studying fundamental groups ⓘ studying homology of covering spaces ⓘ |
| usedToConstruct | presentation matrices for Alexander modules ⓘ |
| uses | formal derivatives ⓘ |
| yearIntroduced | 1953 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.