Patterns in three dimensions

Counting maps from knot groups to solvable groups is hard, already for S₄.

From John Cullen’s knotlab guide

Explore the idea

Three nontrivial character slots

Take the Heisenberg group over the seven-element field, acted on by a cyclic group of order three. Write ω = 2 modulo 7, so ω³ = 1. Choose the action’s first two characters; their product determines the central one.

Character αCharacter β

nontrivial

αβ

ω²

nontrivial

All three slots participate: #P-hard

The twisted obstruction has the three-way interaction that Borromean surgeries can steer into the counting reduction.

For example, α = ω and β = ω² are individually nontrivial, but αβ = 1. Having two nontrivial slots is not enough.

For this family, the note’s theorem places exact counting on the hard side precisely when α, β and αβ are all nontrivial. The controls compute the character values; the complexity classification comes from the proof.

Spaces and their groups. A 3-manifold is a space that looks like ordinary three-dimensional space near each of its points, such as the space around a knotted rope. Its fundamental group records its loops: two loops count as the same when one can be slid into the other, and two loops combine by running one after the other. A homomorphism from the fundamental group to a finite group labels every loop with an element of the group so that running two loops one after the other multiplies their labels. For the space around a knot drawn as a diagram, such a labelling is a colouring of the diagram's arcs that obeys one rule at each crossing. A group is solvable when it is built in finitely many layers from commutative groups. S₃ is the group of the six permutations of three things, and S₄ that of the 24 permutations of four. A group's Fitting height counts its layers of a particular kind, the nilpotent ones, which are the groups closest to commutative; Heisenberg groups are the simplest nilpotent groups that are not commutative. A link is several closed loops of rope, possibly tangled with one another, and a knot is a link with one loop.

What the note proves. When the laws counted are group laws on a 3-manifold, counting worlds means counting homomorphisms from its fundamental group to a finite group. Rowell conjectured in 2009 that this count is computable in polynomial time for solvable groups. Proved (note): unless every #P function is computable in polynomial time, the conjecture fails. On links it fails already for S₃. On knots it fails for S₄, the smallest group for which it can, and for some groups of Fitting height two. For Heisenberg-type groups the note proves an exact dividing line. See notes/knot-group-counting.

Surgery and characters. Two more terms appear in the reading below. Surgery on a link cuts out a thin tube around it and glues the tube back differently, which makes a new 3-manifold. A character of a commutative group maps it to the complex numbers of absolute value 1, preserving multiplication.

Reading. In the Heisenberg families, hardness appears exactly when the twisted obstruction joins three nontrivial characters at once, which three-component, pairwise-unlinked (Borromean) surgeries supply. Everything pairwise stays countable. This echoes the K2 pattern of lesson 2.1, but it is a statement about counting complexity, not about knots in our sense. The knot test cannot tell hard groups from easy ones: every group law, read as the relation x·y = z, is constrained as a whole and free in every pair.