Edges of the contact polytope of the Leech lattice

Question

Let $P\subset\Bbb R^{24}$ be the contact polytope of the Leech lattice, that is, $P$ is the convex hull of the 196,560 shortest vectors of $\Lambda_{24}$.

Question: What are the edges of $P$?

Let's say the norm of any vertex is $4\sqrt 2$. I am pretty sure that any two vertices at distance $4\sqrt 2$ form an edge. But what about vertices farther apart? If I am not mistaken, then for any specific vertex, there are further vertices at distance $8$ and $4\sqrt 6$ (and, of course, the antipodal vertex at distance $8\sqrt2$, but there is no edge obviously)?

I am aware of

• Sikirić, Mathieu Dutour, Achill Schürmann, and Frank Vallentin. The contact polytope of the Leech lattice.

but it only discusses the facets. In fact, I found no other publication discussing this polytope.

Answer 1

Using the unimodular scaling of the Leech lattice, the length of each minimal vector is $\sqrt{4}$. Fixing a particular minimal vector $u$, the remaining minimal vectors $v$ are:

• 1 vector $v$ with $\langle u, v \rangle = 4$ (namely $v = u$);
• 4600 vectors $v$ with $\langle u, v \rangle = 2$;
• 47104 vectors $v$ with $\langle u, v \rangle = 1$;
• 93150 vectors $v$ with $\langle u, v \rangle = 0$;
• 47104 vectors $v$ with $\langle u, v \rangle = -1$;
• 4600 vectors $v$ with $\langle u, v \rangle = -2$;
• 1 vector $v$ with $\langle u, v \rangle = -4$ (namely $v = -u$);

and the Conway group $Co_2$ (the pointwise stabiliser of $\{0, u\}$) acts transitively on each of the seven sets of vertices mentioned above.

Now, suppose that $u, v, x, y$ are distinct minimal-norm lattice vectors, and further suppose that we have:

$$ \lambda u + (1 - \lambda)v = \mu x + (1 - \mu) y $$

where $0 < \lambda, \mu < 1$. Then the line segment with endpoints $u,v$ intersects the line segment with endpoints $x,y$, so neither of these can be edges of the contact polytope $P$.

The special case where $\lambda = \mu = \frac{1}{2}$ is particularly helpful: it states (when you double both sides of the equation) that if $u + v = x + y$, then neither of the line segments are edges. So if $u, v$ does form an edge, then all vectors of the form $x + y$ (where $x, y$ is an arbitrary pair of minimal-length vectors satisfying $\langle x, y \rangle = \langle u, v \rangle$) must be pairwise distinct. But the Leech lattice is additively closed, and we know the number of vectors of each norm, so we can apply the pigeonhole principle:

• There are $\frac{1}{2} \times 196560 \times 1$ sums of pairs of antipodal minimal vectors, but only one Leech vector of length 0, so there are no edges between antipodal vertices.
• There are $\frac{1}{2} \times 196560 \times 4600 = 452088000$ sums of pairs of minimal vectors $u, v$ with $\langle u, v \rangle = -2$, but only $196560$ Leech vectors of length $\sqrt{4}$, so there are no edges of this form.
• There are $\frac{1}{2} \times 196560 \times 47104 = 4629381120$ sums of pairs of minimal vectors $u, v$ with $\langle u, v \rangle = -1$, but only $16773120$ Leech vectors of length $\sqrt{6}$, so there are no edges of this form.
• There are $\frac{1}{2} \times 196560 \times 93150 = 9154782000$ sums of pairs of minimal vectors $u, v$ with $\langle u, v \rangle = 0$, but only $398034000$ Leech vectors of length $\sqrt{8}$, so there are no edges of this form.

On the other hand, this stops working for any closer pairs of vectors:

• There are $\frac{1}{2} \times 196560 \times 47104 = 4629381120$ sums of pairs of minimal vectors $u, v$ with $\langle u, v \rangle = 1$, which is exactly the same as the number of length-$\sqrt{10}$ vectors in the lattice! Assuming that $Co_0$ acts transitively on these vectors (this has probably been proved somewhere?), it means that they're all uniquely expressible as the sum of two minimal vectors.

So we've ruled out edges between pairs of vertices $u,v$ when $\langle u, v \rangle \leq 0$. The closest pairs of vertices clearly form edges, so we know that there's an edge whenever $\langle u, v \rangle = 2$ (i.e. the angle is 60 degrees).

This leaves the delicate case of $\langle u, v \rangle = 1$, where $u - v$ is a vector of length $\sqrt{6}$. To show that these do indeed form an edge, it suffices to show that there are no other vertices $w$ with $\langle w, u + v \rangle \geq \langle u, u + v \rangle = 5$; that way, the midpoint $\frac{1}{2}(u + v)$ cannot be expressed as a combination of any vertices other than $u$ and $v$ (because all other vertices lie on the same side of the perpendicular bisecting hyperplane of $u + v$ and $0$).

Now, if $\langle w, u + v \rangle \geq 5$, then one of $\langle w, u \rangle$ or $\langle w, v \rangle$ must be at least $\frac{5}{2}$. But the inner product of two distinct minimal Leech vectors is at most $2$, so this is only possible if $w = u$ or $w = v$. This proves the desired result, and also implies unconditionally that every length-$\sqrt{10}$ Leech lattice vector is uniquely expressible as the sum of two length-$\sqrt{4}$ lattice vectors.

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In conclusion, $P$ has $196560$ vertices, $452088000$ 'short' edges of length $\sqrt{4}$, and $4629381120$ 'long' edges of length $\sqrt{6}$.