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We characterize the $(1,1)$ knots in the 3-sphere and lens spaces that admit non-trivial L-space surgeries. As a corollary, 1-bridge braids in these manifolds admit non-trivial L-space surgeries. We also recover a characterization of the Berge manifold among 1-bridge braid exteriors.

For every smooth Jordan curve γ and cyclic quadrilateral Q in the Euclidean plane, we show that there exists an orientation-preserving similarity taking the vertices of Q to γ. The proof relies on the theorem of Polterovich and Viterbo that an embedded Lagrangian torus in C2 has minimum Maslov number 2.

We determine the lens spaces that arise by integer Dehn surgery along a knot in the three-sphere. Specifically, if surgery along a knot produces a lens space, then there exists an equivalent surgery along a Berge knot with the same knot Floer homology groups. This leads to sharp information about the genus of such a knot. The arguments rely on tools from Floer homology and lattice theory. They are primarily combinatorial in nature.

We study the chromatic number of the curve graph of a surface. We show that the chromatic number grows like $k\\log k$ for the graph of separating curves on a surface of Euler characteristic  $-k$ . We also show that the graph of curves that represent a fixed nonzero homology class is uniquely $t$ -colorable, where $t$ denotes its clique number. Together, these results lead to the best known bounds on the chromatic number of the curve graph. We also study variations for arc graphs and obtain exact results for surfaces of low complexity. Our investigation leads to connections with Kneser graphs, the Johnson homomorphism, and hyperbolic geometry.

The d -invariant of an integral, positive definite lattice Λ records the minimal norm of a characteristic covector in each equivalence class . We prove that the 2-isomorphism type of a connected graph is determined by the d -invariant of its lattice of integral flows (or cuts). As an application, we prove that a reduced, alternating link diagram is determined up to mutation by the Heegaard Floer homology of the link’s branched double-cover. Thus, alternating links with homeomorphic branched double-covers are mutants.

Let L⊂S3L \\subset S^3 denote an alternating link and Σ(L)\\Sigma (L) its branched double-cover. We give a short proof of the fact that the fundamental group of Σ(L)\\Sigma (L) admits a left-ordering iff LL is an unlink. This result is originally due to Boyer-Gordon-Watson.

We prove that a special alternating knot does not decompose as a non-trivial band sum. This restricts concordances from special alternating knots, and we conjecture that special alternating knots are ribbon concordance minimal. We verify our conjecture in many cases. This work is motivated by another conjecture of Owens and the second author, which posits that the set of alternating knots is downward closed under ribbon concordance.

We show that the signature of a positive braid link is bounded from below by one-quarter of its first Betti number. This equates to one-half of the optimal bound conjectured by Feller, who previously provided a bound of one-eighth.

We prove that for every smooth Jordan curve \\(\\gamma \\subset \\mathbb{C}\\) and for every set \\(Q \\subset \\mathbb{C}\\) of six concyclic points, there exists a non-constant quadratic polynomial \\(p \\in \\mathbb{C}[z]\\) such that \\(p(Q) \\subset \\gamma\\). The proof relies on a theorem of Fukaya and Irie. We also prove that if \\(Q\\) is the union of the vertex sets of two concyclic regular \\(n\\)-gons, there exists a non-constant polynomial \\(p \\in \\mathbb{C}[z]\\) of degree at most \\(n-1\\) such that \\(p(Q) \\subset \\gamma\\). The proof is based on a computation in Floer homology. These results support a conjecture about which point sets \\(Q \\subset \\mathbb{C}\\) admit a polynomial inscription of a given degree into every smooth Jordan curve \\(\\gamma\\).

We construct a version of Lagrangian Floer homology whose chain complex is generated by the inscriptions of a rectangle into a real analytic Jordan curve. By using its associated spectral invariants, we establish that a rectifiable Jordan curve admits inscriptions of a whole interval of rectangles. In particular, it inscribes a square if the area it encloses is more than half that of a circle of equal diameter.