Seifert Surfaces and Genus - Key Properties

The properties of Seifert surfaces and genus reveal fundamental constraints on knot topology and connect to deep results in 4-dimensional topology.

Theorem

Existence and Uniqueness Properties:

Existence: Every knot admits a Seifert surface (Seifert, 1934)
Non-uniqueness: A knot has infinitely many Seifert surfaces
Genus is well-defined: Minimum genus $g(K)$ is a knot invariant
Genus formula for torus knots: $g(T(p,q)) = \frac{(p-1)(q-1)}{2}$

The existence proof uses Seifert's algorithm. Non-uniqueness follows from stabilization: any Seifert surface $F$ can be modified by adding handles to create higher-genus surfaces for the same knot.

Definition

A knot $K$ is fibered if its complement $S^3 \setminus K$ fibers over $S^1$ with fiber a Seifert surface. Equivalently, there exists a Seifert surface $F$ such that the monodromy $h: F \to F$ generates the fibration.

Properties of fibered knots:

Alexander polynomial is monic: highest and lowest coefficients are $\pm 1$
Genus formula: $g(K) = \frac{1}{2}\deg(\Delta_K(t))$ (equality in Fox-Milnor bound)
All torus knots are fibered

Example

The trefoil $3_1$ is fibered:

Alexander polynomial: $\Delta(t) = t - 1 + t^{-1}$ , degree 2
Genus: $g(3_1) = \frac{1}{2} \cdot 2 = 1$ (matches Fox-Milnor bound)
The minimal genus Seifert surface is exactly the fiber

The figure-eight $4_1$ is fibered:

$\Delta(t) = -t + 3 - t^{-1}$ , degree 2 (after normalization)
$g(4_1) = 1$
Monodromy is a Dehn twist

Not all knots are fibered. The knot $5_2$ has $g(5_2) = 1$ but $\deg(\Delta) < 2$ , so it cannot be fibered.

Theorem

Additivity of Genus: For connected sum of knots: $g(K_1 \# K_2) = g(K_1) + g(K_2)$

This makes genus a homomorphism from the knot monoid (under connected sum) to $(\mathbb{Z}_{\geq 0}, +)$ .

Proof

Given minimal genus Seifert surfaces $F_1$ for $K_1$ and $F_2$ for $K_2$ , the connected sum construction yields a Seifert surface for $K_1 \# K_2$ with genus $g(F_1) + g(F_2)$ , proving $g(K_1 \# K_2) \leq g(K_1) + g(K_2)$ .

The reverse inequality follows from Alexander polynomial multiplicativity and the degree bound: $\deg(\Delta_{K_1 \# K_2}) = \deg(\Delta_{K_1}) + \deg(\Delta_{K_2})$ , so $2g(K_1 \# K_2) \geq 2g(K_1) + 2g(K_2)$ .

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Definition

The slice genus $g_4(K)$ is the minimum genus of any smoothly embedded surface in $D^4$ with boundary $K \subset S^3 = \partial D^4$ .

Relationships:

$g_4(K) \leq g(K)$ always (since any Seifert surface in $S^3$ bounds in $D^4$ via cone)
$g_4(K) \leq u(K)$ (unknotting number bound)
For slice knots: $g_4(K) = 0$
$g_4$ is not additive under connected sum!

Example

The square knot $3_1 \# 3_1^*$ (trefoil plus mirror trefoil):

$g(3_1 \# 3_1^*) = g(3_1) + g(3_1^*) = 1 + 1 = 2$
$g_4(3_1 \# 3_1^*) = 0$ (it's slice!)
This demonstrates $g_4$ is not additive

The slice-ribbon conjecture asks: Is $g_4(K) = 0 \iff K$ is ribbon (admits immersed disk with only ribbon singularities)?

Remark

Computing $g(K)$ exactly is difficult. Known methods:

Upper bounds: Seifert's algorithm, Murasugi sum formula
Lower bounds: Alexander polynomial degree, signature, Ozsvath-Szabo $\tau$ invariant from knot Floer homology
Exact computation: For special families (torus knots, two-bridge knots) using formulas

Modern invariants like knot Floer homology provide sharp genus bounds, sometimes determining $g(K)$ exactly through algebraic computations.

These properties establish genus as a fundamental measure of knot complexity, interacting richly with polynomial invariants, fibering properties, and 4-dimensional topology.