The Jones and Alexander Polynomials - Main Theorem

The existence and uniqueness theorems for polynomial invariants establish their fundamental role in knot theory.

Theorem

Uniqueness of Jones Polynomial: There exists a unique function $V: \{\text{oriented links}\} \to \mathbb{Z}[t^{\pm 1/2}]$ satisfying:

Normalization: $V_{\text{unknot}}(t) = 1$
Skein relation: $t^{-1}V_{L_+}(t) - tV_{L_-}(t) = (t^{1/2} - t^{-1/2})V_{L_0}(t)$

Moreover, $V$ is a knot invariant unchanged under ambient isotopy.

Proof

Existence: Define $V$ via the Kauffman bracket $\langle \cdot \rangle$ and writhe normalization: $V_L(t) = \left((-A^3)^{-w(L)}\langle L \rangle\right)\bigg|_{A=t^{-1/4}}$

The Kauffman bracket satisfies relations ensuring the skein equation holds after substitution. Reidemeister invariance of the normalized bracket guarantees $V$ is well-defined on isotopy classes.

Uniqueness: Suppose $V'$ also satisfies the conditions. For any link $L$ , use the skein relation recursively to reduce to unlinks (disjoint unknots). Since both $V$ and $V'$ satisfy the same skein relation and initial condition, they compute identical values. By induction on crossing number, $V = V'$ for all links.

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Theorem

Fox-Milnor Theorem for Alexander Polynomial: The Alexander polynomial $\Delta_K(t)$ of any knot $K$ satisfies:

$\Delta_K(1) = 1$ (normalization)
$\Delta_K(t) = \Delta_K(t^{-1})$ (symmetry)
$\Delta_K(t)$ is a Laurent polynomial in $\mathbb{Z}[t^{\pm 1}]$

Conversely, not every polynomial satisfying these properties is the Alexander polynomial of some knot. The Torres conditions provide additional constraints.

Example

The polynomial $f(t) = t^2 - 5t + 6 - 5t^{-1} + t^{-2}$ satisfies:

$f(1) = 1 - 5 + 6 - 5 + 1 = -2 \neq 1$ (fails normalization)

Even $g(t) = 2 - t - t^{-1}$ satisfies $g(1) = 0 \neq 1$ . The normalization is essential: actual Alexander polynomials must have $\Delta(1) = 1$ by topological necessity (the infinite cyclic cover is connected).

Theorem

Span Inequality for Jones Polynomial: For any knot $K$ , $\text{span}(V_K) \leq 2c(K)$ where $\text{span}(V_K) = \max \deg(V_K) - \min \deg(V_K)$ and $c(K)$ is crossing number.

For alternating knots, equality holds: $\text{span}(V_K) = 2c(K)$ .

This result (Kauffman-Murasugi-Thistlethwaite) resolved Tait's conjecture that reduced alternating diagrams have minimal crossing number.

Remark

The span equality for alternating knots means the Jones polynomial immediately computes crossing number for alternating knots. For non-alternating knots, span provides only a lower bound, and computing exact crossing number remains computationally difficult.

The proof uses the state sum expansion of the Kauffman bracket and properties of alternating link diagrams, particularly that reduced alternating diagrams are "checkerboard colorable" in a specific way.

Theorem

Alexander Polynomial and Genus (Fox-Milnor): For any knot $K$ with genus $g(K)$ , $\deg(\Delta_K) \leq 2g(K)$

This provides a lower bound on genus: $g(K) \geq \frac{1}{2}\deg(\Delta_K)$ .

For fibered knots (knots whose complements fiber over $S^1$ ), equality holds.

Example

The trefoil has $\Delta_{3_1}(t) = t - 1 + t^{-1}$ , so $\deg(\Delta) = 2$ . This gives $g(3_1) \geq 1$ . Since the trefoil is fibered (as a $(2,3)$ -torus knot), we have $g(3_1) = 1$ exactly.

For the $(3,4)$ -torus knot: $\deg(\Delta) = 12$ , giving $g(T(3,4)) \geq 6$ . Since it's fibered, $g(T(3,4)) = 6$ exactly. This matches the formula $g(T(p,q)) = \frac{(p-1)(q-1)}{2}$ for torus knots.

These theorems establish polynomial invariants as powerful tools with rigorous algebraic foundations, connecting combinatorial diagram properties to deep topological invariants like genus and crossing number.