Reidemeister Moves and Invariants - Applications

Reidemeister moves enable practical algorithms and deep theoretical results across mathematics and physics.

Theorem

Unknot Recognition is in NP ∩ co-NP: Given a knot diagram with $n$ crossings:

NP: A certificate that the diagram is the unknot is a sequence of Reidemeister moves to the standard unknot diagram (verifiable in polynomial time)
co-NP: A certificate that the diagram is not the unknot is a knot invariant value differing from the unknot

Hass-Lagarias-Pippenger (1999) showed unknot recognition is actually in P, though their algorithm requires $2^{cn}$ time for some large $c$ , making it impractical.

Modern practical algorithms use normal surface theory, Floer homology, and hyperbolic geometry heuristics, often solving real instances in seconds despite worst-case exponential complexity.

Theorem

Minimal Crossing Number is Additive on Connected Sums (Schubert, 1954): For knots $K_1, K_2$ : $c(K_1 \# K_2) = c(K_1) + c(K_2)$

This was proven using Reidemeister move analysis on Seifert surfaces. The result shows crossing number behaves like a "logarithm" under connected sum, analogous to how logarithm converts multiplication to addition.

Example

Knot Tabulation: Creating complete tables of prime knots by crossing number uses Reidemeister moves systematically:

Generate all possible diagram configurations with $n$ crossings
Apply Reidemeister moves to find canonical forms
Eliminate duplicates using invariants (Jones, Alexander, etc.)
Verify primality and completeness

Current tables extend to 19 crossings (millions of distinct knots), with computational methods handling the combinatorial explosion. The number of prime knots with $n$ crossings grows exponentially: approximately $\frac{1}{n}(\frac{7}{e})^n$ knots asymptotically.

Theorem

Thistlethwaite's Theorem (1985): The Jones polynomial $V(t)$ can distinguish the unknot from all other knots when combined with signature and Alexander polynomial.

More precisely: If $V(K) = 1$ , $\sigma(K) = 0$ , and $\Delta(K) = 1$ , then $K$ is the unknot.

This application of Reidemeister-invariant polynomials provided the first algorithmic unknot recognition using quantum invariants.

Example

Quantum Computing Applications: Topological quantum computation uses knot theory and braid group representations. Reidemeister moves correspond to:

Type I: Single-qubit gates (phase gates)
Type II: Two-qubit gates (CNOT-type operations)
Type III: Quantum circuit equivalences (Yang-Baxter equations)

The Yang-Baxter equation from statistical mechanics is exactly the algebraic expression of Reidemeister move III in the braid group, creating deep connections between knot theory and quantum information.

Theorem

Witten's Invariants and Chern-Simons Theory: The Jones polynomial and its generalizations arise from 3D Chern-Simons gauge theory. The path integral formulation automatically satisfies Reidemeister move invariance because: $Z(K) = \int \mathcal{D}A \, \exp\left(i k \int_{M} \text{Tr}(A \wedge dA + \frac{2}{3} A \wedge A \wedge A)\right)$

is manifestly invariant under diffeomorphisms of the 3-manifold $M = S^3$ .

This physics perspective shows Reidemeister moves emerge naturally from gauge symmetry, providing non-perturbative definitions of knot invariants.

Example

DNA Topology and Molecular Biology: DNA strands form knots and links during replication and recombination. Enzymes called topoisomerases change DNA topology by:

Type I topoisomerases: Cut one strand (analogous to Reidemeister I and II)
Type II topoisomerases: Pass one strand through another (analogous to crossing changes)

Modeling these processes requires tracking knot types through Reidemeister moves. For instance, bacteriophage P4 DNA forms trefoil knots, and understanding recombination pathways uses knot theory to predict product distributions.

Theorem

Virtual Knot Theory Applications: Virtual Reidemeister moves (adding move IV for virtual crossings) extend classical theory. Virtual knots:

Model knots in thickened surfaces
Arise in Gauss diagrams and combinatorial knot theory
Connect to free knots and welded knots

Virtual knot invariants (like the virtual Jones polynomial) solve some classical problems. For instance, Kauffman's proof that the virtual Kishino knot is nontrivial uses techniques unavailable in classical knot theory.

Remark

The universality of Reidemeister moves—appearing in quantum physics, molecular biology, quantum computation, and pure topology—demonstrates how fundamental combinatorial structures underlie diverse phenomena. This universality stems from the moves capturing the essence of dimensionality reduction: projecting 3D topology to 2D diagrams while preserving equivalence relations.

These applications show Reidemeister moves transcend their original purpose, becoming a universal language for topological equivalence across mathematics, physics, and biology.