Research
Preprints
A slope p/q is characterising for a knot K if the oriented homeomorphism type of the manifold obtained by Dehn surgery of slope p/q on K uniquely determines the knot K. We combine analysis of JSJ decompositions with techniques involving lengths of shortest geodesics to find explicit conditions for a slope to be characterising for K in the case where K is any hyperbolic knot or any satellite knot by a hyperbolic pattern. Assuming that the list of 2-cusped orientable hyperbolic 3-manifolds obtained using the computer programme SnapPea is complete up to a certain point, we use hyperbolic volume inequalities to generate a refinement for the special case of Whitehead doubles. We also construct pairs of multiclasped Whitehead doubles of double twist knots for which 1/q is a non-characterising slope.
Projects
Characterising slopes for satellite knots of hyperbolic type
Joint with Patricia Sorya.
For any knot K of hyperbolic type (whose complement has a hyperbolic outermost JSJ piece), we have realised the bound C(K) such that |q|≥C(K) implies that p/q is a characterising slope for K.
Quasiquadruple grid diagrams for Lagrangian surfaces
Joint with Shintaro Fushida-Hardy and Devashi Gulati.
Analogously to triple grid diagrams encoding Lagrangian surfaces in $\mathbb{CP}^2$, we introduce a new notion of quadruple grid diagrams encoding Lagrangian surfaces in S^2 x S^2. We investigate how to reconcile the geometry with the combinatorics and give some examples. We are looking for applications relating to embedded Lagrangian surfaces in S^2 x S^2!
This project began at the Summer Trisectors Workshop (2022).
Sliceness of Whitehead doubles of (2, 2k+1)-torus knots
Joint with Charles Stine.
We use RBG links to construct a knot with the same 0-trace as a satellite knot which is concordant to the Whitehead double of a (2, 2k+1)-torus knot. This has potential applications to open questions about sliceness.
Realising any slope as non-characterising
Joint with Kyle Hayden and Lisa Piccirillo.
Given any slope p/q, can we find two knots K and K' which share homeomorphic p/q-surgeries? Under investigation!
Other
PhD thesis
A hyperbolic perspective on the Dehn surgery characterisation problem
Please note: there are some mistakes in the final chapter. Contact me if you would like to access the corrected version.