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Preprints
Preprints
Characterising slopes for hyperbolic knots and Whitehead doubles
Characterising slopes for hyperbolic knots and Whitehead doubles
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.
Effective bounds on characterising slopes for all knots
Effective bounds on characterising slopes for all knots
Joint with Patricia Sorya.
We combine new applications of results from hyperbolic geometry with previous individual work of the authors to determine, for any given knot K, an explicit bound C(K) such that |q| > C(K) implies that p/q is a characterising slope for K.
Dehn surgery functions are never injective
Dehn surgery functions are never injective
Joint with Kyle Hayden and Lisa Piccirillo.
We prove that, for each fixed rational number p/q, there exists a pair of distinct knots whose p/q-surgeries are orientation-preservingly homeomorphic. This confirms a 1978 conjecture of Gordon.
Projects
Projects
Quasiquadruple grid diagrams for Lagrangian surfaces
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 notion of quasiquadruple grid diagrams encoding Lagrangian surfaces in S^2 x S^2 (and further generalisations). We construct non-orientable Lagrangian surfaces in S^2 x S^2.
This project began at the Summer Trisectors Workshop (2022).
Knotted spheres via the RBG construction
Knotted spheres via the RBG construction
Joint with Valentina Bais, Alessio Di Prisa, Daniel Hartman, Chun-Sheng Hsueh, Marc Kegel, Alice Merz, Mark Pencovitch, Arunima Ray, Diego Santoro and Paula Truöl.
We generalise the RBG link construction to construct two non-isotopic 2-knots K and K' which share diffeomorphic 5-dimensional traces. Conversely, we show that the unknot is detected by its trace in higher dimensions.
Sliceness of Whitehead doubles of (2, 2k+1)-torus knots
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.
Smoothly non-characterising slopes which do not give contactomorphic surgeries
Smoothly non-characterising slopes which do not give contactomorphic surgeries
Joint with Tanushree Shah.
We investigate whether smoothly non-characterising slopes can be characterising in the contact setting.
Other
Other
PhD thesis
PhD thesis
A hyperbolic perspective on the Dehn surgery characterisation problem
Please note: there are some mistakes in the final chapter. The new and improved version is essentially my joint paper with Patricia Sorya.
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