Life is easy. If you find it to be hard, it means you’re looking at it in the wrong way.– Tomer, Kathmandu (Israeli TV series).
Can the same be said of mathematics? In mathematics, it’s a common occurance to run up against a problem which one finds to be difficult. But then it’s often the case that a logically equivalent problem in a different model is much easier.
In Low-Dimensional Topology, particularly in Knot Theory, we have many competing models for our objects, and we can ask logically equivalent questions in each one. I don’t know if any one of these categories is inherently complexity-theoretically easier than any other- as far as I know it may all depend on the specific context. But I’ve recently been reading Mark Powell’s thesis A second order algebraic knot concordance group, in which he puts forward a new model for knots which seems well-suited to a certain sort of problem which I care about, and which I am excited about.
First, as an aside, to give one characteristic example of a problem (which has nothing to do with topology) being easy in one model but hard in another, let’s say that we wanted to prove that the sum of two solutions to Pell’s Equation ( for a nonsquare integer) is again a solution to Pell’s equation. It would be difficult to prove this over . But instead, one can consider Pell’s Equation over the real numbers . Proving the assertion for integers in is logically equivalent to proving it over , but is complexity-theoretically easier (evidence: Gödel Speedup Theorem), and we can prove it without much trouble by noting that the (positive branch of the) curve has parametric equations , , and is therefore isomorphic to under addition. Letting , , the sum of points and is seen to be , and the sum of integer points is seen to be an integer point. Indeed, the complexity-theoretical improvement from to may perhaps be considered the rationale for defining the real numbers in the first place, and in particular for Calculus. (This philosophical introduction draws upon answers to this MO question.)
Back to Knot Theory, before I discuss symmetric Poincaré triads, let me recall for comparison some other knot models that we have (some more similar than others). Of course there are more. Too many, if you ask me.
- Smooth embeddings of a circle in (or in ).
- PL embeddings of a circle in (or in ).
- Smooth or PL 3-manifolds , or the 3-manifold you get by 0-surgery on .
- Geometrized 3-manifolds with a cusp.
- Knot diagrams in or in modulo Reidemeister moves.
- Gauss diagrams.
- Elements of planar algebras.
- Groups with peripheral structure.
- Braids modulo Markov moves.
- Closures of tangles.
- 2-jets of functions from the circle to the real line.
Powell suggests the following model, which is inspired by High-Dimensional Knot Theory, particularly by the work of his supervisor Andrew Ranicki. First, let us set up the geometric model which we would like to recapture. A knot complement is a 3-manifold with a torus boundary that is marked by a meridian and longitude . From the perspective of knot concordance, the fundamental operation on knot complements is the connect-sum. So split the boundary into two copies of along two meridians (splitting the logitude into two segments) .
To connect-sum, throw away one from each knot, identify meridians, and plug what’s left, to obtain a new knot complement. This idea in itself is already lovely- I had never before seen the connect-sum operation formulated in a way that descends to the level of chain complexes.
Here now is the symmetric Poincaré triad of a knot. Trade the knot exterior , the two copies of , and for chain complexes with coefficients in . Poincaré duality is the key property of chain complexes in Knot Theory from which many proofs stem (e.g. The Fox-Milnor proof that the Alexander polynomial of a slice knot factors as .) So each chain complex should come equipped with a symmetric structure, that is a collection of maps from the chain complex to the dual cochain complex that capture Poincaré duality on the level of complexes, and which descends to the well-known Poincaré-Lefshetz duality isomorphisms upon passing to homology, modulo the relevant equivalence relation.
Then, a symmetric Poincaré triad of a knot, AKA a knot, is a diagram of chain complexes with symmetric structure, maps between them, and a homotopy between the maps to make the diagram commute:
It takes Powell 70 pages to explain the construction of a symmetric Poincaré triad of a knot, but in a field where terseness is the norm, I would say that it is 70 pages well spent. It’s a lucid and enjoyable construction, and reading it taught me a lot.
You can connect-sum symmetric Poincaré triads I think (Powell only does it modulo the third commutator subgroup of the fundamental group), and when you’re finished, you have a bona fide algebraic model for knots in the context of knot concordance. What’s so attractive about this model is that it’s easy to manipulate via bordism modulo an algebraic equivalence relation, via Ranicki’s algebraic theory of surgery- messy Kirby calculus considerations are replaced by the localization exact sequence. Symmetric Poincaré triads are a pain to build but a joy to work with.
Algebra is generous; she often gives more than is asked of her. -Jean le Rond d’Alembert.