Let $a,b$ be to simple closed loops on a surface $S$ with homologically trivial intersection (more generally I'm also interested in the case when $b$ is 1-codimensional). Denote their intersection on $a$ orientation-wise by $x_1,\cdots,x_r$, with orientations $or_{x_i}\in \{\pm 1\}$. I'm interested in the amplitude of $\mathbb N \to \mathbb Z, i \mapsto \sum_{j\leq i} or_{x_j}$.

Is this somehow a studied quantity? Is there a relation with the geometric intersection number? Probably not, so is there a connection between homotopy representants of $a$ which minimize 1) the above quanitity and 2) the geometric intersection?


  • $\begingroup$ What do you mean by "amplitude"? $\endgroup$ Feb 8, 2016 at 18:18
  • 1
    $\begingroup$ I didn't really know terminology for what I wanted to say, so I actually borrowed this word from music theory. It basically means the difference between the maximal and minimal value. Or if $\Gamma$ is the graph, I mean the order of the image $\Gamma \to \mathbb N \times \mathbb Z \to \mathbb Z$. $\endgroup$ Feb 8, 2016 at 18:22
  • $\begingroup$ It seems like that number depends on the ordering of the $x_i$ (unless I'm confused about something), so I'm not sure how natural it is. $\endgroup$ Feb 8, 2016 at 20:56
  • $\begingroup$ Actually you're right, I was silently assuming trivial algebraic intersection number. But it should be natural then (or I'm confused about something). $\endgroup$ Feb 9, 2016 at 11:11


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