Research

My research interests include asking questions about Lefschetz fibrations. Donaldson and Gompf’s combined results show an interesting correspondence between symplectic 4-manifolds and Lefschetz pencils (which can be blown up to be a Lefschetz fibration). I like to ask questions about these manifolds because they are often reduced to questions about the mapping class group, braids, and branched covers. Often, I study properties of genus-2 Lefschetz fibrations (which have the extra structure of being hyperelliptic) and then I generalize properties to all genera fibers. See some slides (with plenty of pictures!) of some of my previous talks below:

Click here for slides to the Making Science Accessible Seminar Talk

Click here for slides to the U Alabama Topology & Algebra Seminar Talk

Papers & Publications:

S. Knavel. Another proof of Matsumoto’s Relations of Lefschetz fibrations. In prep.

S. Knavel, R. Van Why. Compactifications of Stein domains and Lefschetz fibrations. In prep.

S. Knavel, T. Rodewald. On Lagrangian Cobordisms and the Chekanov-Eliashberg DGA. October 2025. <click here to read preprint>

S. Knavel. A note on the topology of Lefschetz fibrations. October 2025. arXiv:2510.03173

S. Knavel, R Trapp. Embedded totally geodesic surfaces in fully augmented links. Communications in Analysis and Geometry Volume 31, Number 3, 563-593, 2023 <link>

R. Bycofski, B. Klein, and S. Knavel. (2020, Release Jan 2.). The Pancake Problem. In, D. Auckly, B. Klein, A. Serenevy, and T. Shubin (Eds.) Inspiring Mathematics: Lessons from the Navajo Nation Math Circles. Jointly published by the American Mathematical Society (AMS) and the Mathematical Sciences Research Institute (MSRI) as Volume 24 of the Math Circles Library. bookstore.ams.org

S. A. Knavel, T. V. Savin, M. V. Mroz, M. E. Kordesch, C. N. Eads, J. T. Sadowski, and S. A. Tenney. A Mathematical Model of Solid-State Dewetting of Barium Thin Films on W(112). Mathematical Modelling of Natural Phenomena. EDP Sciences, 2019.

Invited Talks & Seminars:

  • (Upcoming) 2 plus 2 equals 4: A look into 4-dimensional symplectic topology. November 10, 2025. Making Science Accessible Seminar: Communicating Science to the Public. College of Sciences, Georgia Tech

  • On symplectic divisors in Lefschetz fibrations, October 8, 2025, University of Alabama Topology Seminar <link>

  • New first Betti number bounds on Lefschetz fibrations, September 26, 2025, University of Massachusetts Amherst, 5 Colleges Geometry and Topology Seminar <link>

  • Genus-2 Lefschetz Fibrations, May 16-18, 2025, Co-organizer of “Women and Gender Minorities in Symplectic and Contact Geometry and Topology (WISCons) in Madison,” 2025 AWM Research Symposium, Madison, WI <link>

  • Genus-2 Lefschetz Fibrations, May 3-4, 2025, AMS Spring Western Sectional Meeting, California Polytechnic State University, San Luis Obispo CA <link>

  • Betti Numbers and Indecomposability of Genus-2 Lefschetz Fibrations. Claremont Topology Seminar, April 15, 2025, Claremont Center for Mathematical Sciences, Claremont CA<link to abstract>

  • Genus-2 Lefschetz Fibrations, Mar 8-9, 2025, Spring Southeastern AMS Sectional Meeting, Clemson University, Clemson SC <link to abstract>

  • Genus-2 Lefschetz Fibrations, Oct 16 2024, Vanderbilt Topology Seminar <link to abstract>

  • cancelled due to weather (rescheduled for March 28-9, 2026): Title TBD. 2024 Fall Southeastern AMS Sectional Meeting, Georgia Southern University, Savannah GA

  • “Lefschetz fibrations with Abelian  fundamental groups,” 2024 Spring Southeastern Sectional AMS Meeting, Florida State University, Tallahassee, FL <link to abstract>

  • “Lefschetz fibrations with fixed finitely presented fundamental groups,” Joint Mathematics Meeting 2024, San Francisco, CA <link to abstract>

  • “Fundamental Groups of genus-2 Lefschetz fibrations,” Dec 2023, Tech Topology Summer School <link>

  • “A Lefschetz fibration construction with arbitrary finitely presented fundamental group,” July 2023, Tech Topology Summer School <link>

Internal Seminars & Talks:

  • “Two-fold branched covers of hyperelliptic Lefschetz fibrations,” Feb 2024, <link to abstract>

  • “Genus-2 Lefschetz fibrations,” Oct 2023, <link to abstract>

  • “An introduction to 4-manifolds,” Sep 2023, <link to abstract>

  • “An exotic contractible 4 manifold,” Sep 2022, <link to abstract>

  • “Constructing Exotic 4-manifolds via cork twists, Gluck twists, and log transforms,” Dec 2021, <link to abstract>

Other Conferences, Service, and Attendances:

  • JMM 2026 Wiscons Special Session Organizer (with Sumeyra Sakalli & Clair Dai)
  • Graduate student TA for PUMA: Practicum for Undergraduate MAthematicians – Algebraic and Computational Topology, Apr 12-13, 2025, University of California, Los Angeles <link>
  • AWM 2025 Research Symposium Organizing Committee <link>
  • 2025 Georgia International Topology Conference <link>
  • JMM 2025 <link>