Advanced Topics

A Seifert surface is a connected, orientable surface whose boundary is the knot. Seifert proved in 1934 that every knot has one, and his proof is the algorithm in the demo. First, orient the knot. At each crossing, cut both strands and reconnect the four ends the only way that keeps the arrows consistent. This leaves disjoint oriented Seifert circles. Fill each circle with a disk, stacking nested circles at different heights so the disks never meet. Finally, put the crossings back as half-twisted bands joining the disks. Each band's twist follows its crossing, so the band's edges cross with the correct strand on top.

The surface is orientable because each disk's orientation comes from its circle, and every band joins its two disks so that they agree. That is why gold (the side from which the boundary runs counterclockwise) never meets blue. The surface has one disk per circle and one band per crossing, so its Euler characteristic is χ = s − n. A connected orientable surface with one boundary circle has χ = 1 − 2g, so its genus is g = (n − s + 1)/2.

The genus of a knot is the smallest genus of any Seifert surface it bounds. The algorithm's surface depends on the diagram and need not be the smallest. For a reduced alternating diagram it is minimal, a theorem of Crowell and Murasugi (1958–59). All the table presets are drawn that way. The closed braid of 5₂ uses 6 crossings and 3 circles, so it gives genus 2, but 5₂ has genus 1. Switching one crossing of the trefoil gives a 3-crossing diagram of the unknot, which bounds a disk. Seifert's algorithm still builds a genus-1 surface from it.

The readouts use the Alexander polynomial Δ(t), computed from the diagram's Wirtinger presentation. Half the span of Δ is a lower bound for the knot's genus, so when the surface reaches that bound it is certainly minimal. Every non-trivial knot with at most 10 crossings has Δ ≠ 1. Δ also tells apart all knots up to 7 crossings, except that it cannot see mirror images or tell the granny knot from the square knot. The demo names a knot only when one of those facts settles it. Freehand drawings are made alternating, and the PD code below the demo follows the Knot Atlas convention.

The genus g(K) is the minimal genus among all Seifert surfaces spanning the knot. It measures knot complexity: g(K) = 0 if and only if K is the unknot, and genus is additive under connected sum.

Key insight: The genus connects several invariants. The crossing number gives an upper bound g(K) ≤ (c(K)-1)/2, while the Alexander polynomial degree gives a lower bound. Computing exact genus remains difficult in general.

A knot K is fibered if its complement S³ − K is swept out by a circle's worth of disjoint Seifert surfaces, the pages or fibers, all sharing the knot as their edge. Following the pages once around the circle brings each page back to itself by a map called the monodromy. A fiber is always a minimal-genus Seifert surface.

For torus knots, Milnor's construction (1968) gives the pages exactly. On the 3-sphere |z|² + |w|² = 1, the polynomial f = z^p + w^q vanishes on the (p, q) torus knot, and the pages are the sets where arg f is constant. The demo computes the page arg f = 0 by marching cubes and draws it by stereographic projection. It then turns the pages with the symmetry (z, w) ↦ (e^iqφz, e^ipφw), which multiplies f by e^ipqφ. One full turn is a rotation of order pq, so the monodromy is periodic. Each page has genus (p − 1)(q − 1)/2: 1 for the trefoil, 2 for the cinquefoil and 3 for T(3, 4).

Key insight: Stallings' theorem (1962) gives a purely algebraic test: a knot is fibered if and only if the commutator subgroup of its knot group is finitely generated. A fibered knot's Alexander polynomial must be monic (leading coefficient ±1), so 5₂, with Δ = 2t − 3 + 2t⁻¹, is not fibered. The figure-eight knot is fibered too, but its monodromy is pseudo-Anosov rather than periodic. It is the Anosov map of the punctured torus given by the matrix (2 1; 1 1), and there is no polynomial formula as simple as Milnor's for its pages.

By Thurston's classification, every knot is of exactly one of three types: a torus knot, a satellite knot, or a hyperbolic knot, whose complement admits a complete hyperbolic metric. Remarkably, most knots are hyperbolic, and the hyperbolic volume is a powerful, computable invariant.

Key insight: The figure-eight knot has the smallest hyperbolic volume of any knot (~2.02988). By Mostow rigidity, the hyperbolic structure is unique, so the volume is a well-defined invariant -- though distinct knots (such as mutant pairs) can share the same volume.