I’ve recently been helping Stefan Friedl and Nathan Dunfield with an interesting project looking at the twisted Alexander polynomials of hyperbolic knots, which has now resulted in two papers [2,3], some software , and a number of unanswered questions. I’ve found it all fascinating and have learned a lot of interesting stuff about mathematics (twisted Alexander polynomials, Reidemeister torsion, hyperbolic 3-manifolds, the Mahler measure, etc) and computing (the Sage computer algebra system and the Python programming language).
I plan to look into the details and the background in more depth in the next few posts, but the basic idea is that for any knot we can calculate something called the Alexander polynomial . This is a polynomial expression in a single variable which (for various technical reasons) is well-defined up to multiplication by , but the reason we care about it is that it’s an isotopy invariant: it can often help us decide whether two knots are equivalent or not.
More precisely, if you have two knots and which might be equivalent (that is, one is really just a continuously-deformed copy of the other) then you work out their Alexander polynomials and compare them: if then and aren’t equivalent. (Conversely, though, if then that doesn’t help, since there are many examples of non-equivalent knots which have equivalent Alexander polynomials.) For example, the trefoil knot (usually denoted ) has , but the figure-eight knot has , so straight away we know that these are different knots. (My friend Peter recently found this fact, but not the proof, scribbled on a toilet door.)
On the other hand, the 11–crossing knots and have , so although we happen to know by other methods that they’re different, we can’t tell them apart just by using the Alexander polynomial.
But in addition to sometimes being able to tell different knots apart, the Alexander polynomial also contains some geometric information about the knot: it gives us a (not always optimal) lower bound on the genus of the knot, and also a necessary (but not sufficient) condition on whether the knot is fibred. I’ll explain later what the genus of a knot is, and what it means for a knot to be fibred.
Rather than considering the knot on its own, we need to look at the space surrounding it: on its own a knot is really just a closed loop, a circle, and what makes it interesting is how it’s been embedded in whatever version of 3–dimensional space we happen to be using. In practice, we mostly consider knots embedded in the 3–sphere , which can be described in various ways, but a fairly straightforward way of viewing it is like ordinary 3–space joined up at infinity.
During the 1990s, some mathematicians got interested in a generalised version of , called twisted Alexander polynomials. The idea here is that we modify by incorporating some extra information about the space surrounding the knot. More precisely, we use a representation of the fundamental group of this space. It turns out that if we’re careful and/or lucky about our choice of representation, the twisted Alexander polynomial can often distinguish between knots that the ordinary Alexander polynomial can’t, and can sometimes give us better genus and fibring information too.
Now, due to some really important work by William Thurston during the late 1970s and early 1980s (for which he was awarded a Fields medal in 1982), it turns out that the vast majority of knots (more precisely, all but 32 of the 1701936 knots with 16 or fewer crossings) are hyperbolic. (The exceptions are torus and satellite knots, which are comparatively rare cases.)
What this means is that we can impose a canonical 3–dimensional hyperbolic geometric structure on the complement of our knot (the complement is basically with a –shaped hole cut out of it). And the symmetries of this geometric structure give us a similarly canonical representation which we can use to define a twisted Alexander polynomial with. Stefan’s idea was that since the hyperbolic structure is in some sense god-given, and the corresponding “holonomy” representation is similarly divinely-inspired, these twisted Alexander polynomials (which we called ) should be pretty special, and hopefully have some interesting and useful properties.
So we set our computers to work calculating for all 313209 hyperbolic knots with 15 or fewer crossings, using the Sage computer algebra system and a hyperbolic topology package called SnapPy (a Python library written by Nathan Dunfield and Mark Culler, based on Jeff Weeks‘ SnapPea software). We found that not only does correctly predict the genus in all 313209 cases, it also correctly detects fibredness too.
There’s an operation called mutation you can do on a knot: broadly speaking, you cut out a segment (a tangle) of the knot, flip it over, and glue it back in. Sometimes (the simplest examples have 11 crossings) you get a different knot as a result. The most famous example pair consists of the Conway and Kinoshita–Terasaka knots; the knots and mentioned earlier are another example pair. The ordinary Alexander polynomial can’t tell the difference (we say it’s mutation invariant) but the hyperbolic twisted Alexander polynomial often can. (In particular, it can tell the difference between the Conway and Kinoshita–Terasaka knots, and also between and .)
 N M Dunfield, Genus-Comp, software and documentation (2011)
 N M Dunfield, S K Friedl, N J Jackson, Twisted Alexander polynomials of hyperbolic knots, preprint (2011) arXiv:1108.3045
 S K Friedl, N J Jackson, Approximations to the volume of hyperbolic knots, from: “Twisted topological invariants and topology of low-dimensional manifolds”, (T Morifuji, editor), RIMS Kôkyûroku 1747 (2011) 35–46 arXiv:1102.3742