Why are the Reidemeister moves so important?

Reidemeister proved that two diagrams represent the same knot exactly when one can be obtained from the other by these three local moves, so checking that a quantity is unchanged by them proves it is a genuine invariant.

What is the Seifert genus of a knot?

It is the smallest genus among all orientable surfaces in space whose boundary is the knot; it is an invariant that measures the knot's complexity and is additive under connected sum.

Knot Invariants

A knot invariant is a quantity that does not change when a knot is deformed, providing a tool to prove that two knots are genuinely different.

Definition

A knot invariant is a function on knots that takes equal values on equivalent knots, so that different values certify that two knots are not ambient-isotopic; equivalently, it is any quantity preserved under the three Reidemeister moves.

Scope

This topic covers the principle that any quantity unchanged under the Reidemeister moves is a knot invariant, and surveys the classical invariants: the knot group (the fundamental group of the complement), the Seifert surface and Seifert genus, the crossing number, the unknotting number, bridge number, and tricolorability. It treats Seifert matrices and the signature, the limitations of individual invariants, and the role of invariants in detecting chirality and distinguishing knots that look superficially similar.

Core questions

How do the Reidemeister moves reduce the question of invariance to a finite, checkable condition?
What geometric and algebraic invariants — knot group, genus, signature — capture distinct features of a knot?
Why can an invariant distinguish some knots but fail to separate others?
How do invariants detect properties such as chirality and the unknotting number?

Key concepts

Reidemeister moves and invariance
Knot group and the knot complement
Seifert surfaces, Seifert genus, and Seifert matrix
Crossing, unknotting, and bridge numbers
Signature and tricolorability

Clinical relevance

Knot invariants are what make knot theory applicable: they distinguish DNA topoisomers in molecular biology and provide the obstructions used in classifying three-manifolds via surgery on knots and links.

History

Reidemeister proved in 1927 that his three moves generate knot equivalence, reducing invariance to local checks; Seifert's construction of spanning surfaces (1934) gave the genus and signature, and these classical invariants formed the backbone of the subject before the polynomial era.

Key figures

Kurt Reidemeister
Herbert Seifert
Dale Rolfsen

Seminal works

lickorish1997
rolfsen1976

Frequently asked questions

Why are the Reidemeister moves so important?: Reidemeister proved that two diagrams represent the same knot exactly when one can be obtained from the other by these three local moves, so checking that a quantity is unchanged by them proves it is a genuine invariant.
What is the Seifert genus of a knot?: It is the smallest genus among all orientable surfaces in space whose boundary is the knot; it is an invariant that measures the knot's complexity and is additive under connected sum.