Seifert surfaces, knot cobordism, and hyperbolic knots
You've mastered the fundamentals - now it's time to explore the cutting edge! In this module, we'll dive into Seifert surfaces (orientable surfaces bounded by knots), knot genus (the minimal complexity measure), fibered knots (knots whose complements bundle over the circle), and hyperbolic geometry (Thurston's revolutionary geometric classification).
These advanced topics connect knot theory to differential geometry, dynamical systems, 3-manifold topology, and geometric analysis. They represent some of the most beautiful mathematics of the 20th and 21st centuries, including work by Fields Medalists Thurston and Perelman.
A Seifert surface is an orientable surface whose boundary is the knot. Every knot bounds such a surface! The Seifert algorithm provides a constructive method to build one.
Choose a direction around the trefoil
Smooth all 3 crossings following orientation
Two Seifert circles result from smoothing
Attach a disk to each of the two circles
Add 3 twisted bands at original crossing locations
The Seifert surface of the trefoil is a punctured torus (genus 1).
For a Seifert surface with c Seifert circles and n crossings:
This follows from the Euler characteristic: ฯ = 2 - 2g = c - n for a surface with one boundary component.
| Knot | Crossings (n) | Circles (c) | Genus (g) |
|---|---|---|---|
| Unknot (0โ) | 0 | 1 | 0 |
| Trefoil (3โ) | 3 | 2 | 1 |
| Figure-Eight (4โ) | 4 | 3 | 1 |
Step 1 - Orient: Choose a direction to traverse the knot (clockwise or counterclockwise).
Step 2 - Resolve: At each crossing, smooth it out by following the orientation. This breaks the knot into disjoint circles.
Step 3 - Count circles: The Seifert circles are the connected components after smoothing.
Step 4 - Attach disks: Fill each circle with a flat disk.
Step 5 - Add bands: At each original crossing, add a twisted band to reconnect the disks, forming the surface.
Seifert surfaces are fundamental to knot theory:
The genus g(K) is the minimal genus among all Seifert surfaces spanning the knot. It measures the complexity of the knot - higher genus means more "complicated" topology.
The Seifert algorithm always produces a surface, but it might not be minimal!
Several invariants provide bounds on the genus:
| Knot | Crossings | Genus | Unknotting # | Type |
|---|---|---|---|---|
| Unknot (0โ) | 0 | 0 | 0 | |
| Trefoil (3โ) | 3 | 1 | 1 | Torus knot |
| Figure-Eight (4โ) | 4 | 1 | 1 | Hyperbolic |
| Cinquefoil (5โ) | 5 | 2 | 2 | Torus knot |
| Three-Twist (5โ) | 5 | 2 | 1 | Twist knot |
| Stevedore (6โ) | 6 | 1 | 1 | Hyperbolic |
A knot is slice if it bounds a smooth disk in the 4-ball Bโด. The 4-dimensional genus (slice genus) can be smaller than the 3-dimensional genus!
The genus is a fundamental measure of knot complexity:
A knot K is fibered if its complement Sยณ - K can be expressed as a surface bundle over the circle: Sยณ - K โ F ร Sยน / ~, where F is the fiber (a surface) and ~ identifies boundaries via a homeomorphism called the monodromy.
The genus of the fiber surface F
The surface automorphism ฯ: F โ F
Fibers are once-punctured tori, monodromy is a Dehn twist
The monodromy ฯ: F โ F is classified by the Thurston-Nielsen classification:
Theorem (Stallings, 1962): A knot is fibered if and only if its Alexander polynomial is monic (leading and trailing coefficients are ยฑ1).
Theorem (Torus Knots): All torus knots T(p,q) are fibered with fiber of genus g = (p-1)(q-1)/2.
Theorem (Fiber vs Genus): If a knot is fibered, the fiber is a minimal genus Seifert surface.
Fibered knots have special properties that make them easier to study:
By Thurston's geometrization theorem, every knot complement admits one of three geometries: spherical, Euclidean, or hyperbolic. Remarkably, most knots are hyperbolic!
The figure-eight has the smallest hyperbolic volume of any knot: ~2.02988...
Satellite knots and unknot
Torus knots and cables
Most knots!
| Knot | Geometry | Hyperbolic Volume |
|---|---|---|
| Unknot (0โ) | Spherical | โ |
| Trefoil (3โ) | Euclidean | โ |
| Figure-Eight (4โ) | Hyperbolic | 2.02988 |
| Cinquefoil (5โ) | Euclidean | โ |
| Three-Twist (5โ) | Hyperbolic | 2.82812 |
| Stevedore (6โ) | Hyperbolic | 3.16396 |
William Thurston proved (completed by Perelman) that every closed 3-manifold decomposes into pieces, each with one of 8 geometric structures. For knot complements:
Thurston's geometrization revolutionized 3-dimensional topology:
Herbert Seifert proves every knot bounds an orientable surface and gives a constructive algorithm to build one from any diagram.
John Stallings proves a knot is fibered if and only if its Alexander polynomial is monic. This connects algebra to the geometric fibration structure!
William Thurston revolutionizes 3-manifold topology by proving most knot complements admit hyperbolic geometry. Introduces hyperbolic volume as a complete invariant. Earned the Fields Medal in 1982.
Grigori Perelman completes Thurston's geometrization conjecture using Ricci flow, simultaneously proving the Poincarรฉ conjecture. Awarded (but declined) the Fields Medal in 2006.
The genus is defined as the minimum genus of all Seifert surfaces. The Seifert algorithm gives an upper bound, but finding the minimum is hard!
If a knot is fibered, the fiber is automatically a minimal genus Seifert surface. Fibered knots have the nicest geometric structure!
The Seifert matrix (from a Seifert surface) computes both the Alexander polynomial and the signature. Surfaces provide the bridge to algebra!
Thurston's theorem classifies ALL knots: satellite (rare), torus (finitely many per crossing), or hyperbolic (generic). Geometry determines topology!
The hyperbolic volume should be related to the growth rate of colored Jones polynomials. Proved for some special cases, but still open in general!
Does every slice knot (bounds a disk in 4-ball) also bound a ribbon disk? This connects 3D and 4D topology. Still unsolved!
Is there an algorithm to compute the slice genus (4D genus)? We can compute the 3D genus (sometimes), but the 4D version is much harder!
1. Seifert Surfaces: We learned the algorithmic Seifert construction: orient, resolve crossings into Seifert circles, attach disks, add twisted bands. Every knot bounds an orientable surface! The formula g = (2 + n - c)/2 relates genus to crossings and circles.
2. Knot Genus: We explored the minimal genus invariant g(K), which measures knot complexity. The unknot has g=0, trefoil has g=1, cinquefoil has g=2. The genus satisfies g(K) โค (c(K)-1)/2 and u(K) โค g(K), connecting different invariants.
3. Fibered Knots: We discovered that fibered knots have complements that bundle over Sยน. All torus knots are fibered (Dehn twist monodromy), while many hyperbolic knots are fibered (pseudo-Anosov monodromy). Stallings' theorem: fibered iff monic Alexander polynomial!
4. Hyperbolic Geometry: We explored Thurston's revolutionary geometrization: most knots are hyperbolic! The hyperbolic volume is a complete invariant. Figure-eight has minimal volume โ 2.02988. Geometry determines topology!
Next Up: You've now mastered the core theory! Ready to see applications? In the next phase, you'll discover how knot theory connects to DNA biology, quantum computing, statistical mechanics, and materials science. Mathematics meets the real world!