Knots, Links, and Diagrams - Main Theorem

The fundamental theorem connecting abstract knots to their diagrammatic representations forms the foundation of computational knot theory.

Theorem

Reidemeister's Theorem (1927): Two knot diagrams represent equivalent knots if and only if one can be transformed into the other by a finite sequence of:

Type I (Twist): Add or remove a twist in a strand
Type II (Poke): Add or remove two crossings where one strand passes over another
Type III (Slide): Move a strand over or under a crossing

along with planar isotopies (continuous deformations in the plane).

This theorem is remarkable because it reduces the infinite-dimensional problem of ambient isotopy in 3-space to a combinatorial problem involving three local moves on 2-dimensional diagrams.

Remark

The proof has two parts:

(Sufficiency) Each Reidemeister move can be realized by an ambient isotopy of the knot in $\mathbb{R}^3$ , so diagrams related by these moves represent equivalent knots.
(Necessity) Any ambient isotopy between knots can be decomposed into generic moves in a suitable projection, and these project to sequences of Reidemeister moves plus planar isotopies.

The necessity direction requires careful analysis of critical points during the isotopy and is more technically demanding.

Theorem

Uniqueness of Prime Decomposition: Every knot $K$ admits a unique factorization $K = K_1 \# K_2 \# \cdots \# K_n$ into prime knots, unique up to reordering and taking mirror images.

This result, due to Schubert (1949), makes the monoid of knots under connected sum remarkably similar to the multiplicative monoid of positive integers. The proof uses a delicate argument about the intersection of essential surfaces in the knot complement.

Example

The granny knot $3_1 \# 3_1$ (two right-handed trefoils) and square knot $3_1 \# 3_1^*$ (right-handed trefoil plus left-handed trefoil) are both composite knots with the same prime factors. However:

Granny knot: chiral, has nonzero signature
Square knot: achiral (equal to its mirror), zero signature

This shows how the order and orientation of factors matters in knot composition, distinguishing it from integer multiplication.

Theorem

Unknotting Number Bounds: For any nontrivial knot $K$ with crossing number $c(K)$ , $1 \leq u(K) \leq c(K) - 1$ where $u(K)$ is the minimum number of crossing changes needed to unknot $K$ .

Moreover, $u(K_1 \# K_2) \leq u(K_1) + u(K_2)$ with equality for many knots.

The unknotting number measures the "complexity" of a knot in a different way than crossing number. Computing it remains algorithmically difficult, though modern invariants provide bounds.

Remark

Recent developments using Khovanov homology and knot Floer homology have provided new lower bounds for unknotting number. For instance, the $(2,2k+1)$ -torus knots satisfy $u(T(2,2k+1)) = k$ , proven using signature and Arf invariant arguments.

The four-dimensional analog, the slice genus $g_4(K)$ , satisfies $g_4(K) \leq u(K)$ since each unknotting crossing change can be realized by a band move on a surface in $D^4$ . This connects classical knot invariants to 4-manifold topology.

These foundational theorems establish the combinatorial and algebraic framework for all subsequent work in knot theory, transforming intuitive geometric concepts into rigorous mathematical structures amenable to computation and proof.