# Knots, minimal surfaces and J-holomorphic curves

### APA

Fine, J. (2022). Knots, minimal surfaces and J-holomorphic curves. Perimeter Institute. https://pirsa.org/22030106

### MLA

Fine, Joel. Knots, minimal surfaces and J-holomorphic curves. Perimeter Institute, Mar. 11, 2022, https://pirsa.org/22030106

### BibTex

@misc{ pirsa_22030106, doi = {10.48660/22030106}, url = {https://pirsa.org/22030106}, author = {Fine, Joel}, keywords = {Mathematical physics}, language = {en}, title = {Knots, minimal surfaces and J-holomorphic curves}, publisher = {Perimeter Institute}, year = {2022}, month = {mar}, note = {PIRSA:22030106 see, \url{https://pirsa.org}} }

Joel Fine Université Libre de Bruxelles

## Abstract

Let K be a knot or link in the 3-sphere. I will explain how one can count minimal surfaces in hyperbolic 4-space which have ideal boundary equal to K, and in this way obtain a link invariant. In other words the number of minimal surfaces doesn’t depend on the isotopy class of the link. These counts of minimal surfaces can be organised into a two-variable polynomial which is perhaps a known polynomial invariant of the link, such as HOMFLYPT. “Counting minimal surfaces” needs to be interpreted carefully here, similar to how Gromov-Witten invariants “count” J-holomorphic curves. Indeed I will explain how these minimal surface invariants can be seen as Gromov-Witten invariants for the twistor space of hyperbolic 4-space. Whilst Gromov-Witten theory suggests the overall strategy for defining the minimal surface link-invariant, there are significant differences in how to actually implement it. This is because the geometry of both hyperbolic space and its twistor space become singular at infinity. As a consequence, the PDEs involved (both the minimal surface equation and J-holomorphic curve equation) are degenerate rather than elliptic at the boundary. I will try and explain how to overcome these complications.

Zoom link: https://pitp.zoom.us/j/95847819111?pwd=MVg5dWFsNUpiZWVPL1l4Uk9PV2tZZz09