Reidemeister Moves and Invariants - Key Properties

The structure and properties of Reidemeister moves reveal fundamental constraints on knot invariants and diagram complexity.

Definition

The Reidemeister number $R(K)$ of a knot $K$ is the minimum number of Reidemeister moves needed to transform any two minimal diagrams of $K$ into each other. Computing $R(K)$ is generally intractable, but upper bounds exist for specific knot families.

For a diagram transformation, we track:

$R_I$ : number of Type I moves
$R_{II}$ : number of Type II moves
$R_{III}$ : number of Type III moves
Total: $R = R_I + R_{II} + R_{III}$

Remark

Hass and Lagarias (2001) showed that for a knot $K$ with crossing number $c(K) = n$ , any two minimal diagrams can be connected by a sequence of at most $2^{cn}$ Reidemeister moves for some constant $c$ . This exponential bound reflects the computational complexity of the unknotting problem.

More surprisingly, certain knots require Type III moves even when transforming between diagrams with the same number of crossings, showing Type III is genuinely necessary despite preserving crossing number.

Definition

The Reidemeister spectrum describes how invariants change under each move type:

Type I invariant: unchanged under R-I (e.g., most polynomial invariants)
Type II invariant: unchanged under R-II (e.g., writhe is not)
Type III invariant: unchanged under R-III (nearly all interesting quantities)

A quantity is a knot invariant if and only if it's invariant under all three types plus planar isotopies.

Example

Consider the writhe $w(D)$ of an oriented diagram $D$ : $w(D) = \sum_{\text{crossings}} \text{sign}(c_i)$

where sign $(c_i) = +1$ for positive crossings and $-1$ for negative crossings.

Under Type I: $w$ changes by $\pm 1$ (adds/removes twist)
Under Type II: $w$ is preserved (adds $+1$ and $-1$ crossings)
Under Type III: $w$ is preserved (redistributes crossings)

The writhe is not a knot invariant, but it becomes one after normalization in the Jones polynomial.

Theorem

Invariance Criterion: A diagram function $f(D)$ is a knot invariant if and only if:

$f(D) = f(D')$ whenever $D$ and $D'$ are related by planar isotopy
$f(D_{\text{before R-I}}) = f(D_{\text{after R-I}})$ for all Type I moves
$f(D_{\text{before R-II}}) = f(D_{\text{after R-II}})$ for all Type II moves
$f(D_{\text{before R-III}}) = f(D_{\text{after R-III}})$ for all Type III moves

This criterion is the standard method for verifying that polynomials like Jones, Alexander, and HOMFLY-PT are genuine knot invariants.

Example

The Kauffman bracket satisfies:

Type I: NOT invariant (multiplication by $-A^3$ )
Type II: Invariant (follows from skein relation)
Type III: Invariant (requires calculation)

To obtain a genuine invariant, we normalize: $X(K) = (-A^3)^{-w(D)} \langle D \rangle$ . The writhe term exactly cancels the Type I non-invariance, producing the Jones polynomial $V(t) = X(K)|_{A = t^{-1/4}}$ .

Remark

Modern computer algebra systems implement Reidemeister move verification automatically. Given a proposed knot invariant defined by algebraic rules, software can systematically check all move types symbolically, proving invariance without case-by-case geometric arguments.

The computational complexity of minimizing Reidemeister sequences connects to the broader question of whether unknot recognition is in polynomial time (solved affirmatively by Hass-Lagarias-Pippenger, though their algorithm is not practical).

Definition

A virtual knot generalizes classical knots by allowing virtual crossings (crossings in diagrams that don't represent actual over/under information). Virtual knots have four Reidemeister moves (three classical plus one for virtual crossings) and arise naturally in the study of knots in thickened surfaces.

Virtual knot theory shows that classical knots are special: they're virtual knots that can be represented without virtual crossings. Many invariants extend naturally to the virtual setting, providing new perspectives on classical problems.