On the number of embedded essential surfaces in cusped 3-manifolds

Speaker: Jessica Purcell (Monash University)

Time: 2025-11-26 12:00-13:30
Zoom Meeting ID: 994 0480 0043 Passcode: 955282

Abstract:

It is natural to ask how many isotopy classes of embedded essential surfaces lie in a given 3-manifold. The first bounds on the number of such surfaces were exponential, using normal surfaces. More recently, by restricting to alternating link complements in 3-sphere, Hass, Thompson and Tsvietkova obtained polynomial bounds, but for a limited class of surfaces: closed and spanning ones. In this talk, we discuss how to complete the picture for classical alternating links, and how to extend these results to other classes of cusped 3-manifolds. We give explicit polynomial bounds on all embedded essential surfaces, closed or any boundary slope, orientable or non-orientable. Our 3-manifolds are complements of links with alternating diagrams on wide classes of surfaces in broad families of 3-manifolds. This includes all alternating links in 3-sphere as well as many non-alternating ones, alternating virtual knots, many toroidally alternating knots, and most Dehn fillings of such manifolds.

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About Speaker:

Jessica Purcell is a Professor of Mathematics at Monash University. Jessica’s research is in geometry and topology. She works on problems in the overlap of hyperbolic geometry, 3-manifolds, and knot theory. She obtained a PhD in Mathematics from Stanford University, held postdoctoral positions at the University of Texas in Austin and the University of Oxford.