The Jones and Alexander Polynomials - Key Properties

The structural properties of knot polynomials reveal deep algebraic and topological relationships.

Theorem

Fundamental Properties of Jones Polynomial:

Normalization: $V_{\text{unknot}}(t) = 1$
Writhe independence: Properly normalized via $(-A^3)^{-w(D)}$ factor
Bracket span: For alternating knots, $\max(\deg V) - \min(\deg V) = c(K)$
Chirality detection: $V_K(t) \neq V_{K^*}(t^{-1})$ for chiral knots
Unknot detection (conjectured): $V_K(t) = 1 \implies K = \text{unknot}$

The bracket span property for alternating knots, proven by Kauffman, Murasugi, and Thistlethwaite, resolved Tait's 100-year-old conjectures about alternating diagrams. This demonstrates how modern polynomial invariants solve classical geometric problems.

Definition

The Alexander module $A_K = H_1(\tilde{X}_K; \mathbb{Z})$ is the first homology of the infinite cyclic cover $\tilde{X}_K$ of the knot complement $X_K = S^3 \setminus K$ . This module has a $\mathbb{Z}[t^{\pm 1}]$ -module structure induced by deck transformations.

The Alexander polynomial is the order of this module: $\Delta_K(t) = \det(tV - V^T)$ where $V$ is any Seifert matrix for $K$ .

Example

Torus knot $T(p,q)$ with $p, q$ coprime: $\Delta_{T(p,q)}(t) = \frac{(t^{pq} - 1)(t - 1)}{(t^p - 1)(t^q - 1)}$

For $T(2,3)$ (trefoil): $\Delta(t) = \frac{(t^6-1)(t-1)}{(t^2-1)(t^3-1)} = t - 1 + t^{-1}$

For $T(3,4)$ : $\Delta(t) = \frac{(t^{12}-1)(t-1)}{(t^3-1)(t^4-1)}$

The Alexander polynomial of torus knots has this beautiful closed form, contrasting with the more complex Jones polynomial expressions.

Remark

Comparative strengths:

Alexander polynomial: Easier to compute via Seifert matrices; detects sliceness obstructions; related to homology and covering spaces; fails to detect chirality
Jones polynomial: Stronger invariant distinguishing more knots; detects chirality; connected to quantum physics; harder to compute for large knots
Both fail: Neither detects the unknot perfectly (conjectured but unproven); both assign the same value to Kinoshita-Terasaka and Conway knots

Theorem

Torres Conditions: For any knot $K$ , the coefficients of $\Delta_K(t)$ alternate in sign when $\Delta_K(t)$ is written in the form: $\Delta_K(t) = c_0 + c_1(t + t^{-1}) + c_2(t^2 + t^{-2}) + \cdots$

This follows from the geometric interpretation via Seifert surfaces and provides constraints on which polynomials can be Alexander polynomials of knots.

Example

The determinant $\det(K) = |\Delta_K(-1)|$ is an integer invariant. For any knot, $\det(K)$ is always odd. For the trefoil: $\det(3_1) = |\Delta_{3_1}(-1)| = |-1 - 1 - 1| = 3$

The determinant counts $\mathbb{Z}_2$ colorings and equals the linking number in the double branched cover. It provides a simple but useful invariant computable from either Alexander or Jones polynomials.

These properties establish polynomial invariants as central tools in modern knot theory, bridging classical topology, modern algebra, and quantum physics.