| Résume | Let L be an oriented link in the 3-sphere. We will think of the 3-sphere as the ideal boundary of hyperbolic 4-space H^4. My talk will be about trying to count the number of oriented minimal surfaces in H^4 which have L as their ideal boundary. Firstly, I conjecture that this is a topological invariant of the link L. In other words, the number of minimal fillings doesn’t change under isotopies of the ideal boundary. Secondly, I conjecture that the number of genus g fillings of a link with k components gives the coefficient of z^{2g-1+k} in the Alexander polynomial of L. I will explain why I believe these conjectures, why I find them exciting, what has been proved so far, and what remains to be done. If there is time I will also discuss a version of the conjecture relating the HOMFLYPT polynomial to counts of minimal surfaces. |
