Sebastian Baader - On the crossing number of curves on surfaces

Leading question: How many double points do $N$ simple closed non-isotopic curves on a standard surface on genus $g$ have.

Standard surface: closed connected orientable smooth surface. Double point: Intersection between a pair of to curves. We are interested in the lowest possible number of double points.: $cr(N,g)=min\{\sum _{k<l}i(c_k,c_l):c_1,...,c_N\}$ where the curves on the end are simple closed pairwise non-isotopic, essential curves.

We write $cr(3,1)$ for the lowest number of double points of three curves on the torus. The answer is $3$. $cr(7,2)=6$. i.e. $6$ curves on the genus $2$ surface.

Generally speaking this is not an easy problem, especially when a lot of curves are involved.

First equations

This is an example from Teichmüller theory. $cr(3g-3,g)=0$ We draw a cylinder with handles attached. The curves given are the follows:

• Curves around the meridian of the handles
• Curves separating the handle from the main body
• Curves between handles.

This corresponds exactly with the pants decomposition. We can add more curves around the longitudes of the handles, giving us $cr(4g-3,g)=g$ since the meridians intersect the longitude once. (This can be made precise relatively easily)

We make it more complicated. We add a new curve with slope $1$ on each handles. Each handle now has three intersections. $cr(5g-3,g)=3g$

Looking at it abstractly, the three equations give us three critical moments. We can add $3g-3$ curves without problems. Then we can add one curve, giving one more intersections every time. This also only goes up to a point after which the crossings increase by $2$.

Let’s generalize the above. What is $cr(kg-3,g)$ with $k\in \mathbb{N}$?

Uniqueness and enlargenment

The curves realizing the curves are generally not unique (up to homeo of ${\Sigma }_g$ and homotopy) but there are soe special configurations which are unique. Is the number of configuration interesting too? Also, not every minimal $N$-system can be enlarged to a minimal $(N+1)$-system (enlarged in a sense that minimizes curve crossings?)

If not unique, does the count of solutions have an interesting invariant? Maybe for knots?

Q. Can every min. $N$-system be reduced to a min (N-1)-system. Jasmin Jörg ‘24 answered this be no. She also found out some uniqueness properties for low $k$. She answered most questions above with a nice example. There is also some kind of connections to sysoles (see Ingrid Irmer - Schmutz-Thurston duality). It is some kind of curve with minimal length.

Q. What is the asymptotic growth rate of $cr(N,g)$ for fixed $g$. Of course it is monotonic but not necessarily strictly, as we showed before. We where able to show $N^2\le cr(N,g)\le N^3$ We can simply show the upper bound by giving class of examples. The bounds are only up to growth, they are missing some concrete information.

Better Bounds are given by Hugo Parlier et al. $N^{2+\frac{1}{6g-g}}\le cr(N,g)N^{2+\frac{1}{3g-3}}$ We draw a picture to show the upper bound.

question: Is the function $cr(N,g)$ somehow close to the above upper bound? By contradiction, we can show that this is wrong. Honestly, pretty smart to check an upper bound. Assume its true and derive impossible properties. From here on, all proof do not rely on pictures anymore but are solved arithmetically instead. We get some interesting connection to the special linear group of finite fields.