The Jones and Alexander Polynomials - Core Definitions

Polynomial invariants revolutionized knot theory by providing computable algebraic tools that distinguish knots through their algebraic properties.

Definition

The Jones polynomial $V_K(t) \in \mathbb{Z}[t^{\pm 1/2}]$ is defined via the Kauffman bracket for oriented knot diagrams. For an oriented link $L$ with writhe $w(L)$ : $V_L(t) = \left((-A^3)^{-w(L)} \langle L \rangle\right)\bigg|_{A = t^{-1/4}}$

Equivalently, $V$ satisfies the skein relation: $t^{-1} V_{L_+}(t) - t V_{L_-}(t) = (t^{1/2} - t^{-1/2}) V_{L_0}(t)$ where $L_+, L_-, L_0$ differ at one crossing, with $V_{\text{unknot}}(t) = 1$ .

Discovered by Vaughan Jones in 1984, this polynomial was the first new knot invariant in over 60 years, sparking a revolution in knot theory and connecting it to quantum field theory, statistical mechanics, and von Neumann algebras.

Definition

The Alexander polynomial $\Delta_K(t) \in \mathbb{Z}[t^{\pm 1}]$ is the oldest knot polynomial invariant, discovered in 1928. It can be defined via:

Seifert surface approach: From Seifert matrix $V$ , $\Delta_K(t) = \det(tV - V^T)$
Skein relation: $\Delta_{L_+}(t) - \Delta_{L_-}(t) = (t^{1/2} - t^{-1/2}) \Delta_{L_0}(t)$ with $\Delta_{\text{unknot}}(t) = 1$
Homology: As the order of the first homology of the infinite cyclic cover of $S^3 \setminus K$

Example

Trefoil knot $3_1$ :

Jones: $V_{3_1}(t) = t + t^3 - t^4$
Alexander: $\Delta_{3_1}(t) = t - 1 + t^{-1}$

Figure-eight knot $4_1$ :

Jones: $V_{4_1}(t) = t^2 - t + 1 - t^{-1} + t^{-2}$
Alexander: $\Delta_{4_1}(t) = -t + 3 - t^{-1}$

Unknot:

Both: $V(t) = \Delta(t) = 1$

The Jones polynomial detects the chirality of $3_1$ since $V_{3_1^*}(t) = t^{-1} + t^{-3} - t^{-4} \neq V_{3_1}(t)$ , while Alexander cannot: $\Delta_{3_1}(t) = \Delta_{3_1^*}(t)$ .

Remark

Key properties distinguishing these polynomials:

Alexander: Symmetric, $\Delta_K(t) = \Delta_K(t^{-1})$ ; cannot detect chirality; related to homology
Jones: Generally asymmetric; detects chirality; related to quantum groups and statistical mechanics
Alexander: Multiplicative under connected sum: $\Delta_{K_1 \# K_2} = \Delta_{K_1} \cdot \Delta_{K_2}$
Jones: No simple product formula for connected sums

Both satisfy $\Delta_K(1) = 1$ and $V_K(1) = 1$ for all knots.

Definition

The HOMFLY-PT polynomial $P(a, z) \in \mathbb{Z}[a^{\pm 1}, z^{\pm 1}]$ generalizes both Jones and Alexander via the skein relation: $a P(L_+) - a^{-1} P(L_-) = z P(L_0)$ with $P(\text{unknot}) = 1$ .

Setting $a = t^{-1}$ and $z = t^{-1/2} - t^{1/2}$ recovers Jones polynomial. Setting $a = 1$ and $z = t^{1/2} - t^{-1/2}$ recovers Alexander polynomial (up to normalization).

The HOMFLY-PT polynomial provides a two-variable framework encompassing multiple classical invariants, revealing their common algebraic structure through quantum group theory and the representation theory of $\mathfrak{sl}(N)$ .