research

(w/ A. Wand) Stein-fillability and positivity in the mapping class group. 2024. arXiv

• We construct an infinite family of non-positive open books with once-punctured torus pages that support Stein-fillable contact structures. Combined with a result of Wendl, this allows us to give a complete answer to a long-standing question about the mapping class group of a compact surface with boundary: namely, we conclude that the monoid of monodromies supporting Stein-fillable contact structures is equal to the monoid of positive monodromies if and only if the surface is planar.

(w/ P. Feller) The twisting number of a ribbon knot is bounded below by its doubly slice genus. 2024. arXiv

• The twisting number of a ribbon knot K is the minimal number of tangle replacements on the symmetry axis of J#−J for any knot J that is required to produce a symmetric union diagram of K. We prove that the twisting number is bounded below by the doubly slice genus and produce examples of ribbon knots with arbitrarily high twisting number, addressing a problem of Tanaka. To this end, we establish a result of independent interest that the doubly slice genus of K is bounded above by the oriented band move distance between K and any weakly doubly slice link, which also enables us to determine hitherto unknown doubly slice genera of some knots with 12 crossings.

(w/ F. C. Köse) On the symmetric braid index of ribbon knots. 2024. Accepted for publication in Math. Proc. Camb. Phil. Soc. arXiv

• We define the symmetric braid index b_s(K) of a ribbon knot K to be the smallest index of a braid whose closure yields a symmetric union diagram of K, and derive a Khovanov-homological characterisation of knots with b_s(K) at most three. As applications, we show that there exist knots whose symmetric braid index is strictly greater than the braid index, and deduce that every chiral slice knot with determinant one has braid index at least four. We also calculate bounds for b_s(K) for prime ribbon knots with at most 11 crossings.

(w/ J. Simone) On chain link surgeries bounding rational homology balls and χ-slice 3-braid closures. 2024. Accepted for publication in Tran. AMS. arXiv

(w/ A. Wand) Stein-fillable open books of genus one that do not admit positive factorisations. Math. Res. Lett. 30.3 (2023), pp. 709–719; arXiv

On slice alternating 3-braid closures. Pacific J. Math. 324.1 (2023), pp. 49–70; arXiv

• • We construct ribbon surfaces of Euler characteristic one for several infinite families of alternating 3-braid closures. We also use a twisted Alexander polynomial obstruction to conclude the classification of smoothly slice knots which are closures of alternating 3-braids with up to 20 crossings.
  • Accompanying SageMath scripts:
    • K1: Jupyter worksheet, HTML version
    • K2: Jupyter worksheet, HTML version
    • K3: Jupyter worksheet, HTML version

(w/ Z. Dancso, L. Ferretti, S. Friedl, T. Hirsch, C. Johnson, T. Kitayama, J. Lanier, S. Tillmann) Invariants of spatial graphs. In: MATRIX-MFO Tandem Workshop: Invariants in Low-Dimensional Topology: Combinatorics, Geometry, and Computation. 21.3 (2025), pp. 2517–2520; Oberwolfach Reports