Self-intersection numbers of curves on the punctured torus

TL;DR

This paper establishes bounds on the minimal self-intersection numbers of curves on the punctured torus based on their algebraic description length, providing explicit classifications and computational data.

Contribution

It introduces explicit bounds and classifications for self-intersection numbers of curves on the punctured torus, combining combinatorial, topological, and computational methods.

Findings

Bounds on self-intersection numbers in terms of algebraic length

Explicit classification of classes attaining bounds

Computational data for classes up to length 12

Abstract

The minimum number of self-intersection points for members of a free homotopy class of curves on the punctured torus is bounded above in terms of the number L of letters required for a minimal description of the class in terms of the generators of the fundamental group and their inverses: it is less than or equal to (L-2)^2/4 if L is even, and (L-1)(L-3)/4 if L is odd. The classes attaining this bound are explicitly described in terms of the generators; there are (L-2)^2 + 4 of them if L is even, and 2(L-1)(L-3) + 8 if L is odd; similar descriptions and totals are given for classes with self-intersection number equal to one less than the maximum. Proofs use both combinatorial calculations and topological operations on representative curves. Computer-generated data are tabulated counting, for each non-negative integer, how many length-L classes have that self-intersection number, for each length L less than or equal to 12. Experimental data are also presented for the pair-of-pants surface.