Reidemeister Moves and Invariants - Examples and Constructions

Practical applications of Reidemeister moves demonstrate their power in proving knot equivalences and computing invariants.

Example

Unknotting the Trefoil is Impossible: To prove the trefoil $3_1$ is nontrivial, we must show no sequence of Reidemeister moves transforms it to the unknot.

Direct approach: Try all possible move sequences (infeasible).

Invariant approach: Find an invariant differing on $3_1$ and unknot:

Tricolorability: $3_1$ admits 3 non-trivial colorings, unknot admits 0
Jones polynomial: $V_{3_1}(t) = t + t^3 - t^4 \neq 1 = V_{\text{unknot}}(t)$
Knot group: $\pi_1(S^3 \setminus 3_1)$ is non-abelian, $\pi_1(S^3 \setminus \text{unknot}) \cong \mathbb{Z}$

Each invariant provides an independent proof that no Reidemeister sequence can unknot the trefoil.

Definition

A Reidemeister tangle is a diagram fragment with four boundary points that transforms under generalized Reidemeister moves. Tangles provide modular building blocks for knots:

Rational tangles: built from integer tangles via continued fractions
Algebraic tangles: satisfy polynomial equations
Montesinos tangles: generalize rational tangles

Tangle operations commute with Reidemeister moves, enabling compositional knot analysis.

Example

Proving Knot Equivalence: Show that two different diagrams of the figure-eight knot represent the same knot using explicit Reidemeister moves.

Start with the standard 4-crossing diagram and the alternative "twisted" 6-crossing diagram:

Apply Type I to reduce twists (6 → 5 crossings)
Apply Type II to remove crossing pairs (5 → 4 crossings)
Apply Type III to rearrange into standard form (maintain 4 crossings)
Planar isotopy to match exactly

This explicit sequence proves equivalence constructively. For complex knots, such sequences can involve hundreds of moves.

Remark

The unknot recognition problem—determining if a diagram represents the unknot—is decidable but computationally expensive. Algorithms include:

Haken's normal surface theory (exponential time)
Hass-Lagarias-Pippenger (polynomial time but impractical)
Modern heuristics using hyperbolic geometry and Floer homology

In practice, computing strong invariants (Jones, Alexander, hyperbolic volume) often suffices to prove non-triviality, though proving a suspicious diagram is the unknot remains challenging.

Example

Constructing Invariants via Reidemeister Relations: The HOMFLY-PT polynomial $P(a,z)$ satisfies the skein relation: $a P(L_+) - a^{-1} P(L_-) = z P(L_0)$

where $L_+, L_-, L_0$ differ at one crossing. To verify this defines a knot invariant, check:

Type I: The normalization factor for framing ensures invariance Type II: Adding/removing a crossing pair satisfies the skein relation identically Type III: Direct calculation shows $P$ unchanged, using the relation algebraically

This construction paradigm—define via skein relations, verify via Reidemeister moves—produces the major polynomial invariants.

Definition

Reidemeister coloring: A stronger version of tricolorability using modules over $\mathbb{Z}[t^{\pm 1}]$ . At each crossing, assign elements $a, b, c$ satisfying: $tb - (1-t)a - c = 0$

The set of colorings forms a module whose rank is a knot invariant, related to the Alexander module and polynomial.

Example

Computing Jones Polynomial: For the trefoil $3_1$ with positive crossings: $\langle 3_1 \rangle = A \langle \text{hopf link} \rangle + A^{-1} \langle \text{3 circles} \rangle$

Continuing the recursion: $= A[A\langle \text{unknot} \rangle + A^{-1}\langle \text{2 circles} \rangle] + A^{-1}[(-A^2-A^{-2})^2]$ $= A^2 + A^{-2}(-A^2-A^{-2}) + A^{-1}(A^4 + 2 + A^{-4})$

After simplification and writhe normalization ( $w(3_1) = 3$ ), we obtain: $V_{3_1}(t) = t + t^3 - t^4$

This algorithmic computation works for any diagram, always producing the same value for equivalent knots thanks to Reidemeister invariance.

The power of Reidemeister moves lies in reducing infinite-dimensional topological problems to finite combinatorial ones, making knot theory computationally accessible while maintaining rigorous topological foundations.