Research
I study 3- and 4-manifolds, and their submanifolds. I especially like braids, fibered knots, and branched cyclic covers. I often think about how to study the aspects of the L-space conjecture - Heegaard Floer homology, foliations, and orderability for branched covers.
Preprints/Papers
The Dehn twist coefficient for big and small mapping class groups.
with P. Feller and D. Hubbard, Aug. 2023. Arxiv.We give a new characterization of the fractional Dehn twist coefficient which allows us to generalize the invariant to mapping class groups of infinite-type surfaces. We show that for these so-called "big" mapping class groups, the invariant no longer must be fractional but can achieve any real number. We propose its utility in studying big mapping classes and their geometry.
On χ-slice pretzel links.
with S. Fanelle, E. Huang, B. Huenemann, W. Shen, and J. Simone, Jun. 2023. Arxiv.We investigate a generalization of sliceness for links called χ-sliceness; that is, whether a link bounds a smooth, orientable, possibly disconnected surface with Euler characteristic 1. We study this problem for 4-strand pretzel links for which we can give a classification of χ-sliceness unless the determinant of the link is small. The techniques involve hands-on constructions of surfaces, and Donaldson's diagonalization theorem to obstruct surfaces from existing.
Unknotting via null-homologous twists and multi-twists.
with S. Allen, K. Ince, S. Kim, and B. M. Ruppik; preprint, Nov. 2022. Arxiv.We study relationship between unknotting operations such as crossing changes, null-homologous twists, and multi-twists. We show in particular that the number of null-homologous twists necessary to unknot a knot K is at most thrice the number of multi-twists necessary to unknot K via Kirby moves.
Links all of whose branched cyclic covers are L-spaces.
with A. Issa; Bulletin of the London Mathematical Society. Arxiv. Published.We construct new examples of knots and links whose associated 3-manifolds called n-fold branched cyclic coverings are always L-spaces. To do this, we show that our links have special 2-fold symmetries; using this we express their n-fold cyclic branched coverings as 2-fold coverings branched along two-fold quasi-alternating knots. This guarantees that the manifolds are L-spaces.
Braids, fibered knots, and concordance questions.
with D. Hubbard, K. Kawamuro, F.K. Kose, G. Martin, O. Plamenevskaya, K. Raoux and L. Truong; Proceedings Volume of the 2019 Research Collaboration Conference for Women in Symplectic and Contact Geometry and Topology. Arxiv. Published.We study a variety of concordance questions via braids. In particular, we show for quasipositive braids that a function of the fractional Dehn twist coefficient gives a lower bound on the smooth 4-genus and conjecture that this relation holds in general. This was later shown by P. Feller to be the case in his recent paper.
Left-orderability, branched covers and double-twist knots.
Proc. Amer. Math. Soc. 149 (2021), no. 3, 1343–1358. Arxiv. Published.I determined which n-fold cyclic branched coverings of (most) double-twist knots are or are not left-orderable using representations and the Riley polynomial. I showed the techniques can work (with significant effort) on two-bridge knots with multiple twist regions.
My thesis: Symmetries of knots, branched cyclic covers, and L-spaces
link to full textSome results I proved in graduate school only appear in my thesis, including a general statement about Heegaard Floer detecting certain types of periodic symmetries, and using periodic symmetries to obstruct knots from having all L-space branched cyclic covers.
