Optimizing the Upper-Bound Constant for the Crossing Number of Polynomial Curve Systems
Hyungryul Baik
TL;DR
The paper addresses the problem of bounding the crossing number $\mathrm{Cr}(g,m)$ for curve systems of size $m$ on a genus $g$ surface, focusing on the polynomial regime $m\asymp g^{1+\alpha}$. Building on the fibre-surface construction of Baader–Jørg–Parlier (BJP), it relaxes the symmetric parameter choice and solves an entropy-balance optimization to minimize the leading constant in the upper bound. The main result shows that for every $\alpha>0$ and $\varepsilon>0$, $\mathrm{Cr}\bigl(g,\lfloor g^{1+\alpha}\rfloor\bigr) \le (C_\star+\varepsilon)\,\alpha^2\, g^{1+2\alpha}(\log g)^2$ for large $g$, where $C_\star=\inf_{0<x\le 1/2} \frac{2x}{H(x)^2} \approx 1.58054$ and $H(x)=-x\log x-(1-x)\log(1-x)$. This improves the previous constant from $9/4$ by about 30% within the same topological framework by exploiting an entropy–intersection trade-off in the construction of fibre-surface curve systems around complete bipartite graphs. The method combines explicit fibre-surface geometry with Stirling-type entropy bounds and a probabilistic selection argument to realize a large subset of curves with controlled crossings, yielding a sharp constant within BJP’s scheme and clarifying the role of entropy in optimizing such upper bounds.
Abstract
Baader, Jörg, and Parlier recently established an upper bound for the crossing number of curve systems of size $m\asymp g^{1+α}$ on a genus $g$ surface, obtaining a leading coefficient of $9/4=2.25$. Their construction relies on fibre surfaces associated with complete bipartite graphs and uses a symmetric parameter choice corresponding to the central binomial coefficient. In this note, we optimize their construction by relaxing the parameter symmetry and solving the resulting entropy balance problem. We show that for every $α>0$ and every $\varepsilon>0$, \[ \mathrm{Cr}\bigl(g,\lfloor g^{1+α}\rfloor\bigr) \ \le\ (C_\star+\varepsilon)\,α^2\, g^{1+2α}(\log g)^2 \qquad (g\ \text{sufficiently large}), \] where \[ C_\star\ =\ \inf_{0<x\le 1/2}\ \frac{2x}{H(x)^2} \ \approx\ 1.5805443269, \qquad H(x)=-x\log x-(1-x)\log(1-x). \] This reduces the previous constant by about $30\%$ while staying within the same topological framework.