The Jones and Alexander Polynomials - Examples and Constructions

Computing polynomial invariants efficiently requires systematic algorithms exploiting their algebraic structure.

Example

Computing Alexander Polynomial via Seifert Matrix: For the figure-eight knot $4_1$ , construct a Seifert surface from the standard diagram using Seifert's algorithm:

Orient the knot and smooth all crossings
Count Seifert circles: $s = 3$ for $4_1$
Identify crossing bands connecting circles
Build Seifert matrix $V$ encoding linking numbers

For $4_1$ , one Seifert matrix is: $V = \begin{pmatrix} -1 & 1 \\ 0 & -1 \end{pmatrix}$

The Alexander polynomial: $\Delta_{4_1}(t) = \det(tV - V^T) = \det\begin{pmatrix} -t+1 & t \\ -1 & -t+1 \end{pmatrix}$ $= (-t+1)^2 + t = t^2 - 2t + 1 + t = t^2 - t + 1$

Normalizing to $\Delta(1) = 1$ gives $\Delta_{4_1}(t) = -t + 3 - t^{-1}$ in symmetric form.

Definition

Skein tree algorithm for computing Jones polynomial:

Start with oriented diagram $D$ with $n$ crossings
Apply skein relation at one crossing: $V_L = t^{-1}V_{L_+} + tV_{L_-} + \cdots$
Recursively compute for both resolutions
Base cases: unknot ( $V = 1$ ) and unlink ( $V = (-t^{-1/2} - t^{1/2})^{n-1}$ )
Combine using skein relations

Complexity: $O(2^n)$ in worst case, but optimizations reduce this significantly in practice.

Example

Mutant knots: The Conway knot $C$ and Kinoshita-Terasaka knot $KT$ are mutants—they differ by a 180° rotation of a tangle. Mutant knots have identical:

Alexander polynomial: $\Delta_C = \Delta_{KT} = 1$
Jones polynomial: $V_C = V_{KT}$
HOMFLY-PT polynomial
All Vassiliev invariants of degree $< 10$

Yet they are distinct knots! Piccirillo (2020) proved $C$ is not slice while $KT$ is slice, using Khovanov homology to distinguish them. This shows polynomial invariants have fundamental limitations.

Remark

Computational efficiency: Modern software (KnotInfo, SnapPy) uses optimized algorithms:

Gaussian elimination for Alexander via Seifert matrices
Dynamic programming for Jones via skein trees with memoization
Quantum group representations for HOMFLY-PT
Linear algebra over polynomial rings

These enable computing invariants for knots with 15+ crossings in seconds, though asymptotic complexity remains exponential.

Example

Ribbon knots are knots bounding smoothly embedded disks in $D^4$ with only ribbon singularities. For ribbon knots:

Alexander polynomial has form $\Delta_K(t) = f(t)f(t^{-1})$ for some polynomial $f$
Signature $\sigma(K) = 0$

The slice-ribbon conjecture asks if all slice knots are ribbon. The figure-eight $4_1$ is slice (hence ribbon if conjecture holds), consistent with $\Delta_{4_1}(t) = -t + 3 - t^{-1}$ having appropriate form.

Definition

The colored Jones polynomial $J_N(K; q)$ is computed using $N$ -dimensional representations of quantum $\mathfrak{sl}_2$ : $J_N(K; q) = \text{Tr}_N(\rho_N(\beta_K))$ where $\rho_N$ is the $N$ -dimensional irrep and $\beta_K$ is a braid representing $K$ .

For $N=2$ : $J_2(K; q) = V_K(t)|_{t=q^2}$ (standard Jones polynomial)

The colored Jones polynomials contain vastly more information, conjecturally detecting the unknot and encoding hyperbolic volume.

These constructions demonstrate how polynomial invariants transition from abstract algebraic definitions to concrete computational tools, enabling practical knot classification and verification.