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what is your subjective and argumentative structure of Curves in surfaceAre there replacements for the curve complex that make up for its weaknesses?

As far as I know, the most common structure of curves in surface is called Curve the curve complex. John Hempel linked the curve complex and Heegaard Splitting and defined Heeggaard Heegaard Distance. There are lots of results concering on it. You can see about that, e.g., the work of Tseuyoshi Kobayashi, Ruifeng Qiu, Martin Scharlemann, Saul Schleimer, Maggy tomovaTomova, Yair Minsky and so on.But the weak part of these

This structure is has a weak point in that that you can not see any symmetry, and since it is not locally finite, we can not figure out the geodesic. My question is:

Is there any other structure which can avoid the weak part? Or, can you say your favorite structurepoints?Thank you.

Edit: I have changed it. Thanks for advice.
    Post Closed as "subjective and argumentative" by Igor Rivin, S. Sra, Ryan Budney, J.C. Ottem, Mariano Suárez-Alvarez

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what is your favorite subjective and argumentative structure of Curves in surface?

As far as I know, the most common structure of curves in surface called Curve complex. John Hempel linked the curve complex and Heegaard Splitting and defined Heeggaard Distance. There are lots of results concering on it. You can see the work of Tseuyoshi Kobayashi, Ruifeng Qiu, Martin Scharlemann, Saul Schleimer, Maggy tomova, Yair Minsky and so on. But the weak part of these structure is that you can not see any symmetry and since it is not locally finite, we can not figure out the geodesic. My question is
Is there any other structure which can avoid the weak part? Or, can you say your favorite structure? Thank you.

Edit: I have changed it. Thanks for advice.
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