Reidemeister Moves and Invariants - Main Theorem

The completeness of Reidemeister moves establishes them as the foundation for all of diagram-based knot theory.

Theorem

Reidemeister's Theorem (Complete Statement): Two oriented knot diagrams $D_1$ and $D_2$ represent ambient isotopic knots if and only if $D_1$ can be transformed into $D_2$ through a finite sequence of:

Planar isotopies (continuous deformations in the plane)
Type I moves (twist insertions/removals)
Type II moves (crossing pair insertions/removals)
Type III moves (crossing slides)

Moreover, for unoriented diagrams, the same result holds with the addition of orientation reversal.

The "only if" direction is straightforward: each move can be realized by an ambient isotopy in $\mathbb{R}^3$ . The "if" direction requires showing that any continuous deformation of the knot can be approximated by a sequence of diagram moves, a deep result requiring careful analysis of projection critical points.

Proof

(Sufficiency) Each Reidemeister move corresponds to a local ambient isotopy:

Type I: Introduce a small vertical loop pushing strand above/below projection plane
Type II: Push one strand vertically through another
Type III: Perform a local rotation moving one strand past a crossing

Composing these isotopies for a sequence of Reidemeister moves produces an ambient isotopy between the knots represented by the diagrams.

(Necessity - Sketch) Given an ambient isotopy $h_t: S^3 \to S^3$ with $h_0 = \text{id}$ and $h_1(K_1) = K_2$ :

Choose a generic projection direction $v \in S^2$
Perturb $h_t$ to be transverse to projection at almost all times
Critical points occur at finitely many times $t_i$ $t_{i}$ where:
- Crossing is created/destroyed (Type II event)
- Local extremum appears/disappears (Type I event)
- Triple point momentarily occurs (Type III event)
Between critical times, only planar isotopy occurs
Analyze each critical point and show it corresponds to a Reidemeister move

The technical details involve Morse theory and generic position arguments, but the intuition is clear: all possible diagram changes during an isotopy factor through these three local moves.

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Theorem

Markov's Theorem (for Braids and Knots): Two braids $\beta_1 \in B_n$ and $\beta_2 \in B_m$ represent equivalent knots (after closure) if and only if one can be obtained from the other by a sequence of:

Conjugation: $\beta \mapsto \sigma \beta \sigma^{-1}$ for $\sigma \in B_n$
Stabilization: $\beta \in B_n \mapsto \beta \sigma_n^{\pm 1} \in B_{n+1}$

This theorem connects braid group presentations to knot equivalence via a different set of moves.

Remark

Markov's theorem is remarkable because it shows braid equivalence (a purely algebraic group-theoretic problem) captures knot equivalence (a topological problem). This connection enables using braid group representations to construct knot invariants, leading to modern quantum invariants.

Theorem

Completeness of Polynomial Invariants: No finite set of polynomial invariants (Jones, Alexander, HOMFLY-PT, Kauffman) completely classifies knots. There exist infinitely many pairs of distinct knots sharing all these polynomials.

However, the colored Jones polynomials $J_n(K)$ (Jones polynomials computed using representations of quantum $\mathfrak{sl}_2$ ) conjecturally distinguish all knots: $K_1 \cong K_2 \iff J_n(K_1) = J_n(K_2)$ for all $n$ .

This incompleteness motivates categorification: Khovanov homology, which categorifies the Jones polynomial, can sometimes distinguish knots that Jones cannot.

Example

The Kinoshita-Terasaka knot and Conway knot are distinct 11-crossing knots with identical:

Alexander polynomial: $\Delta(t) = 1$ (both have Alexander polynomial of unknot!)
Jones polynomial: $V(t) = t^{-1} - t^{-2} + t^{-3} - t^{-4} + t^{-5}$
HOMFLY-PT and Kauffman polynomials (all equal)

They were finally distinguished in 2020 using Khovanov homology by Piccirillo, who also proved the Conway knot is not slice, resolving a 50-year-old problem.

Theorem

Volume Conjecture (Kashaev, Murakami-Murakami): For a hyperbolic knot $K$ , the colored Jones polynomials encode hyperbolic volume: $\lim_{N \to \infty} \frac{2\pi}{N} \log |J_N(K; e^{2\pi i/N})| = \text{vol}(S^3 \setminus K)$

This connects quantum invariants (polynomial invariants from representations) to classical geometric invariants (hyperbolic volume), one of the deepest conjectures in knot theory.

These theorems establish Reidemeister moves as the algorithmic foundation for knot equivalence while revealing that complete classification requires increasingly sophisticated invariants beyond simple polynomials.