Knots, Links, and Diagrams - Applications

The theoretical foundations of knot diagrams enable powerful applications across mathematics and science.

Theorem

Tait's Conjectures (proven by Kauffman, Murasugi, Thistlethwaite, 1987): For reduced alternating diagrams:

Any reduced alternating diagram of a knot has the minimum possible crossing number
The writhe of any reduced alternating diagram is an invariant of the knot
Reduced alternating diagrams of the same knot have the same number of crossings

These results were open for over 100 years until proven using the Jones polynomial.

The proof elegantly uses the Jones polynomial's relationship to the Kauffman bracket. The key insight is that for alternating knots, the span of the Jones polynomial equals the crossing number, providing a computable invariant that resolves these longstanding conjectures.

Theorem

Wirtinger Presentation: Given a knot diagram $D$ with $n$ crossings and arcs labeled $a_1, \ldots, a_m$ , the fundamental group $\pi_1(S^3 \setminus K)$ has presentation: $\pi_1(S^3 \setminus K) = \langle a_1, \ldots, a_m \mid r_1, \ldots, r_n \rangle$ where each crossing contributes one relation of the form $a_i a_j a_i^{-1} a_k^{-1} = 1$ (with signs depending on crossing type).

Example

For the trefoil knot with standard 3-crossing diagram: $\pi_1(S^3 \setminus 3_1) = \langle a, b \mid aba = bab \rangle \cong \langle x, y \mid x^2 = y^3 \rangle$

This group is non-abelian (unlike $\pi_1(S^3 \setminus \text{unknot}) \cong \mathbb{Z}$ ), proving the trefoil is nontrivial. The abelianization $H_1(S^3 \setminus 3_1) \cong \mathbb{Z}$ shows homology cannot distinguish knots.

Theorem

Link Diagram and Seifert Surfaces: Every knot diagram $D$ gives rise to a canonical Seifert surface via Seifert's algorithm:

Orient the knot and smooth each crossing according to orientation
The resulting disjoint circles (Seifert circles) bound disks
Attach twisted bands at former crossings to form a connected surface

The genus of the resulting surface satisfies $g(D) = \frac{1}{2}(c - s + 1)$ where $c$ is the number of crossings and $s$ is the number of Seifert circles.

Remark

The Seifert surface from this algorithm is not generally minimal genus, but it provides an upper bound: $g(K) \leq g(D)$ for any diagram $D$ of $K$ . For alternating knots, this algorithm often produces minimal genus surfaces.

The Seifert matrix $V$ computed from this surface encodes linking numbers of pushed-off meridians, and its signature gives the knot signature $\sigma(K) = \text{signature}(V + V^T)$ .

Example

Computational Applications: Modern knot theory software (KnotInfo, SnapPy, KLO) uses these theorems to:

Recognize knots from diagrams via polynomial invariants
Compute hyperbolic structures on knot complements
Generate knot tables and verify completeness
Solve topological problems in 3-manifold theory

For instance, SnapPy can determine if a knot complement admits a hyperbolic structure and compute geometric invariants like volume in seconds, using triangulations derived from knot diagrams.

Theorem

Fox $n$ -Colorability: A knot is $n$ -colorable if its diagram can be colored with elements of $\mathbb{Z}_n$ such that at each crossing $c$ with strands $a, b, c$ (where $b$ is the overcrossing), we have $2b \equiv a + c \pmod{n}$ .

The number of distinct $n$ -colorings is an invariant determined by $\det(M)$ where $M$ is the coloring matrix derived from the diagram.

This algebraic invariant generalizes tricolorability and connects to the knot group's representation theory. The determinant $\det(K) = |\det(V + V^T)|$ from the Seifert matrix equals the number of 0-dimensional $SL_2(\mathbb{C})$ representations of $\pi_1(S^3 \setminus K)$ , revealing deep connections to gauge theory and mathematical physics.