Reidemeister Moves and Invariants - Core Definitions

Reidemeister moves are the fundamental local operations on knot diagrams that preserve knot equivalence, forming the bridge between topology and combinatorics.

Definition

The three Reidemeister moves are local transformations on knot diagrams:

Type I (Twist): Add or remove a single twist (kink) in a strand. This changes the writhe by $\pm 1$ and the crossing number by $\pm 1$ .

Type II (Poke): Add or remove a pair of crossings where one strand passes completely over or under another. This changes the crossing number by $\pm 2$ while preserving writhe.

Type III (Slide): Move a strand from one side of a crossing to the other, involving three strands and three crossings. This preserves both writhe and crossing number.

These moves are minimal in the sense that any ambient isotopy of a knot in $\mathbb{R}^3$ can be represented as a sequence of these three local moves on its diagram, combined with planar isotopies. This remarkable fact transforms the continuous problem of knot equivalence into a discrete combinatorial problem.

Definition

A knot invariant is a function $I$ from knots to some set (typically $\mathbb{Z}$ , $\mathbb{Q}$ , or polynomials) such that if $K_1 \cong K_2$ , then $I(K_1) = I(K_2)$ . A diagram invariant remains unchanged under all three Reidemeister moves and planar isotopies.

Classical invariants include:

Crossing number $c(K)$ : minimum crossings in any diagram
Unknotting number $u(K)$ : minimum crossing changes to unknot
Bridge number $b(K)$ : minimum local maxima over all projections
Genus $g(K)$ : minimum genus of spanning Seifert surface

Example

The bracket polynomial $\langle D \rangle$ is defined recursively on diagrams by: $\langle D_+ \rangle = A \langle D_0 \rangle + A^{-1} \langle D_\infty \rangle$ $\langle D \sqcup \bigcirc \rangle = (-A^2 - A^{-2}) \langle D \rangle$ $\langle \bigcirc \rangle = 1$

Here $D_+, D_0, D_\infty$ differ only at one crossing (positive crossing, 0-resolution, $\infty$ -resolution respectively).

The bracket is not a knot invariant because it changes under Type I moves: $\langle D \text{ with twist} \rangle = -A^3 \langle D \rangle$ . However, the normalized version $X(K) = (-A^3)^{-w(D)} \langle D \rangle$ is invariant under all Reidemeister moves, yielding the Kauffman bracket.

Remark

The philosophy behind Reidemeister moves connects to the idea that "knots live in 3D but we study them via 2D projections." The moves encode exactly the freedom we have in choosing projections and continuously deforming the knot in space.

Interestingly, there exist knot diagram invariants (like the Khovanov homology) that are not easily computable from traditional knot invariants, suggesting the diagram perspective carries additional structure.

Definition

An enhanced knot invariant assigns additional structure beyond numbers. Examples include:

Knot polynomials: Alexander $\Delta(t)$ , Jones $V(t)$ , HOMFLY-PT $P(a,z)$
Homology theories: Khovanov homology $Kh(K)$ , knot Floer homology $\widehat{HFK}(K)$
Algebraic invariants: fundamental group $\pi_1(S^3 \setminus K)$ , quandle $Q(K)$
Geometric invariants: hyperbolic volume $\text{vol}(S^3 \setminus K)$

These sophisticated invariants often categorify simpler ones: Khovanov homology categorifies the Jones polynomial, meaning the Jones polynomial arises as the Euler characteristic of Khovanov homology. This hierarchy of invariants reflects deeper categorical structures in knot theory.