Monthly Archives: August 2011

Twisted Alexander polynomials of hyperbolic knots

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 [1], 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 ${K}$ we can calculate something called the Alexander polynomial ${\Delta_K}$. This is a polynomial expression in a single variable ${t}$ which (for various technical reasons) is well-defined up to multiplication by ${\pm t^n}$, 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.

Unexpected cubicle topology

More precisely, if you have two knots ${K_1}$ and ${K_2}$ 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 ${\Delta_{K_1} \not\equiv \Delta_{K_2}}$ then ${K_1}$ and ${K_2}$ aren’t equivalent. (Conversely, though, if ${\Delta_{K_1} \equiv \Delta_{K_2}}$ 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 ${3_1}$) has ${\Delta_{3_1} = t^2 - t + 1}$, but the figure-eight knot ${4_1}$ has ${\Delta_{4_1} = t^2 - 3t + 1}$, 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 ${11_{19}}$ and ${11_{25}}$ have ${\Delta_{11_{19}} = \Delta_{11_{25}} = t^8 - 6t^7 + 18t^6 - 33t^5 + 39t^4 - 33t^3 + 18t^2 - 6t + 1}$, 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 ${S^3}$, which can be described in various ways, but a fairly straightforward way of viewing it is like ordinary 3–space ${\mathbb{R}^3}$ joined up at infinity.

During the 1990s, some mathematicians got interested in a generalised version of ${\Delta_K}$, called twisted Alexander polynomials. The idea here is that we modify ${\Delta_K}$ 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 ${K}$ (the complement is basically ${S^3}$ with a ${K}$–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 ${\mathcal{T}_K}$) should be pretty special, and hopefully have some interesting and useful properties.

So we set our computers to work calculating ${\mathcal{T}_K}$ 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 WeeksSnapPea software). We found that not only does ${\mathcal{T}_K}$ 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 ${11_{19}}$ and ${11_{25}}$ mentioned earlier are another example pair. The ordinary Alexander polynomial ${\Delta_K}$ can’t tell the difference (we say it’s mutation invariant) but the hyperbolic twisted Alexander polynomial ${\mathcal{T}_K}$ often can. (In particular, it can tell the difference between the Conway and Kinoshita–Terasaka knots, and also between ${11_{19}}$ and ${11_{25}}$.)

References

[1] N M Dunfield, Genus-Comp, software and documentation (2011)

[2] N M Dunfield, S K Friedl, N J Jackson, Twisted Alexander polynomials of hyperbolic knots, preprint (2011) arXiv:1108.3045

[3] 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

Three impossible things before teatime

William Blake: The Ancient of Days (1794)

Towards the end of the previous post, I mentioned the infamous 1897 bill considered (and nearly passed) by the Indiana state legislature, which amongst other things declared the value of $\pi$ to be 3.2.  The bill itself mentions three great mathematical factoids discovered by the retired physician and amateur mathematician Dr Edward Johnston Goodwin (c.1828-1902), on which he claims copyright but offers as a gift to the State of Indiana for use in education.  This contribution comprises claimed solutions to three ancient problems unsolved since antiquity: the quadrature of the circle, the trisection of an angle, and the doubling of a cube.  Amongst this discussion, we are presented with the assertion that $\pi=3.2$.

The bill refers to the publication of Goodwin’s work in the American Mathematical Monthly, which is certainly true, although both notes appear in the “Queries and Information” section and are accompanied by the caveat “published by request of the author”.  This latter disclaimer (which doesn’t generally appear in any of the other letters published by the magazine) possibly indicates that the editors were not entirely convinced by Dr Goodwin’s discoveries and wished to make it clear that they weren’t formally endorsing them, whatever Goodwin might have claimed when helping his representative Taylor I Record draft the bill.  (The splendidly-named Record, a farmer and timber merchant, admitted during the debate that he didn’t understand anything of the contents of the bill, but was introducing it at Goodwin’s request.)

So, let’s look at Goodwin’s three discoveries in more detail.

Squaring the circle: This is usually understood to be the problem of constructing, in a finite number of steps, with only a pair of compasses and an unmarked straightedge, a square which has the same area as a given circle.  The problem dates back to the time of the Greek philosopher and mathematician Anaxagoras (c.500-428BC) although Oenopides (c.450BC) is believed to be the first to require only compasses and straightedge; it even puts in an appearance in Aristophanes‘ play The Birds (414BC):

Meton  With the straight ruler I set to work to inscribe a square within this circle; in its center will be the market-place, into which all the straight streets will lead, converging to this center like a star, which, although only orbicular, sends forth its rays in a straight line from all sides.

Goodwin’s method appeared in 1894, in the seventh issue of the first volume [1] and elicited no discussion in later issues (most of the contributing members at that point being embroiled in an ongoing debate over the validity of non-Euclidean geometry), although early in 1896 another article [2] appeared, written by William Heal (member of the London Mathematical Society and Treasurer of Grant County, Indiana), explaining in detail why the problem is unsolvable.  It should be remarked that the problem Heal discusses and Goodwin claims to solve isn’t the classical one: they instead concern themselves with the problem of finding a square with the same perimeter as a given circle.  Where all this falls down has to do with the irrationality, or more specifically the transcendence of $\pi$.

Irrational numbers (roughly, those which cannot be expressed as a fraction) fall into two categories: algebraic numbers like $\sqrt2$ and the golden ratio $\phi=\frac{1+\sqrt5}{2}$ which can be expressed as the roots of polynomials with integer coefficients (respectively, $x^2-2$ and $x^2-x-1$) and transcendental numbers like $\pi$ and $e$, which can’t.  The transcendence of $\pi$ follows from a theorem published in 1882 by the German mathematician Ferdinand von Lindemann (1852-1939), and extended in 1885 by his fellow German Karl Weierstrass (1815-1897).

Goodwin’s article is not exactly a paragon of clear and comprehensible prose, and in places it’s really not obvious either what he’s doing or what he thinks he’s doing.  In fact, as well as the assertion that $\pi=3.2$, he appears to obtain several other values for $\pi$, as discussed by David Singmaster [4] and Arthur Hallerberg [5].

Trisecting an angle: This is another straightedge-and-compasses problem, and as the name suggests, the idea is to construct an angle $\frac\theta3$ from a given angle $\theta$. We can do this for certain angles quite easily (it’s easy to construct an angle of $\frac\pi3$, and also very easy to bisect an angle, so any angle of the form $\frac{\pi}{2^n}$ can be trisected).

The constructability question was solved in general by the French mathematician Pierre Wantzel (1814-1848), in a paper published in 1837 [6].  Wantzel seems to have been a somewhat disorganised and driven person, and a combination of overwork, recourse to unwise levels of artificial stimulants, and generally just not looking after himself, seems to have contributed to his untimely death, a couple of weeks before his 34th birthday.  His near-contemporary Jean-Claude Saint-Venant (1797-1886) remarked later:

He worked usually during the evening, not going to bed until late at night, then reading, and only sleeping poorly for a few hours, alternately abusing coffee and opium; until he married, he took his meals at odd and irregular hours. [7]

The trisection problem is concerned with solving the triple angle formula $\cos(3\theta) = 4\cos^3\theta - 3\cos\theta$.  For example, let’s try to trisect the (constructible) angle $\frac\pi3$, that is, construct the angle $\frac\pi9$.  We know that $\cos\frac\pi3=\frac12$ so our triple angle equation becomes (after a little rearrangement) $8\cos^3\frac\pi9 - 6\cos\frac\pi9 -1 = 0$.  Writing $x=2\cos\frac\pi9$ simplifies this to $x^3-3x-1=0$.  At this point we have to resort to a little bit of Galois theory:

Theorem Let $z_1,\ldots,z_n\in\mathbb{C}$. Any complex number $z$ which is constructible by straightedge and compasses alone from $0,1,z_1,\ldots,z_n$ is algebraic with degree 2 over the field $\mathbb{F} = \mathbb{Q}[z_1,\ldots,z_n,\bar{z}_1,\ldots,\bar{z}_n]$. [8, Section 4.2]

What this means is that in order to construct an angle $\frac\pi9$ with straightedge and compasses, we need to find a quadratic polynomial $ay^2+by+c$ with rational coefficients $a,b,c\in\mathbb{Q}$ which happens to be a factor of our cubic polynomial $x^3-3x-1$.  Equivalently, we need to find a single rational root $x=\alpha\in\mathbb{Q}$ of $x^3-3x-1$.  Now  we make use of another theorem:

Rational Root Theorem Let $p(x) = a_nx^n+\cdots+a_0$ be a polynomial with integer coefficients $a_n,\ldots,a_0\in\mathbb{Z}$.  For any rational solution $x=\pm\frac{p}{q}$, the numerator $p$ must be an integer factor of the constant term $a_0$ and the denominator $q$ must be an integer factor of the leading coefficient $a_n$.

So, the only possible rational roots of our polynomial $x^3-3x-1$ are $\alpha=\pm1$.  But neither of these are roots, since $1^3-3\cdot1-1 = -3$ and $(-1)^3-3(-1)-1 = 1$.  So we can’t reduce our polynomial $x^3-3x-1$ as a product of a linear and a quadratic factor with rational coefficients, and hence by the first theorem above, we can’t construct the number $x=2\cos\frac\pi9$ (or, for that matter, $\cos\frac\pi9$) with just straightedge and compasses.  So there exist perfectly cromulent (even constructible) angles which can’t be trisected with only straightedge and compasses.

Goodwin’s method, by the way, which he described in a very short letter [9] to the American Mathematical Monthly, is to trisect a chord of the given angle:

(A) The trisection of a right line taken as the chord of any arc of a circle trisects the angle of the arc;

This just doesn’t work.  (Try it.)

Doubling a cube:  This third problem fails for very similar reasons to the trisection problem.  Given an arbitrary cube, we want to construct, using only straightedge and compasses, another cube which has exactly twice the volume of the first one.  Equivalently, we need to be able to construct a cube with sides $\sqrt[3]2$ times the length of the original one.  This essentially boils down to a very similar polynomial reduction problem to the previous one, but with the polynomial $x^3-2$ instead of $x^3-3x-1$.  By the same argument, we need to factorise it into a quadratic and a linear factor, both with rational coefficients, and again this really just requires us to find a single rational root.  But by the Rational Root Theorem, such a root must be either $\pm2$.  Which it isn’t, since $2^3-2 = 6$ and $(-2)^3-2 = -10$.  So $x^3-2$ doesn’t factorise into a quadratic and a linear factor with rational coefficients, and hence $\sqrt[3]2$ can’t be constructed with just straightedge and compasses.

Goodwin’s solution [9] was:

(B) Duplication of the Cube: Doubling the dimensions of a cube octuples its contents, and doubling its contents increases its dimensions twenty-five plus one per cent.

This, again, doesn’t work.  What he seems to be saying is that to double the volume of a cube, you need to multiply the length of each side by 1.26.  This is accurate to three decimal places ($\sqrt[3]2 \approx 1.25992105$) but the classical problem requires an exact solution constructible by straightedge and compasses, which this isn’t.

I find cases such as Goodwin’s both interesting and poignant.  He was clearly an intelligent and educated man, and his attempts at these three classical problems seems to have been borne from a genuine desire to advance human knowledge and education.  His obituary, printed in the New Harmony, Indiana Times of 27 June 1902 (quoted in the article by Hallerberg [5]) comments on his sincerity and his ultimate disappointment that his work was not appreciated during his lifetime.

References

[1] E J Goodwin, Quadrature of the Circle, Amer. Math. Monthly 1 (1894) 246-247

[2] W E Heal, Quadrature of the Circle, Amer. Math. Monthly 3 (1896) 41-45

[3] F von Lindemann, Uber die zahl $\pi$, Math. Ann. 20 (1882) 213-225

[4] D Singmaster, The legal values of Pi, Math. Intelligencer 7 (1985) 69-72

[5] A E Hallerberg, Indiana’s squared circle, Math. Mag. 50 (1977) 136-140

[6] P L Wantzel, Recherches sur les moyens de reconnaître si un problème de
Géométrie peut se résoudre avec la règle et le compas
,  J. Math. Pures Appl. (1) 2 (1837) 366-372

[7] J-C Saint-Venant, Biographie: Wantzel, Nouvelles Ann. Math. (1) 7 (1848) 321-331

[8] N Jacobson, Basic Algebra I, W H Freeman (1985)

[9] E J Goodwin, letter, Amer. Math. Monthly 2 (1895) 337