My research interests include asking questions about low-genus Lefschetz fibrations. Donaldson and Gompf’s combined results show a one to one 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 consider a subset of these symplectic 4-manfiolds called hyperelliptic Lefschetz fibrations. They are, in a way, nicer to study because their invariants are possible to compute by hand.
Clemson AMS Talk Slides:
Undergraduate publications:
S. Knavel, R Trapp. Embedded totally geodesic surfaces in fully augmented links. Communications in Analysis and Geometry Volume 23, Number 4, 797–826, 2021 <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: Title tbd, 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
- upcoming: Title tbd, May 3-4, 2025, AMS Spring Western Sectional Meeting, California Polytechnic State University, San Luis Obispo
- upcoming: UCLA Topology Seminar, April 15
- Genus-2 Lefschetz Fibrations, Mar 8-9, 2025, Spring Southeastern AMS Sectional Meeting, Clemson University, Clemson SC <link to abstract>(see slides below)
- Genus-2 Lefschetz Fibrations, Oct 16 2024, Vanderbilt Topology Seminar <link to abstract>
- cancelled due to weather: 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 Wiscon 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>
- Joint Mathematics Meeting 2025 <link>
The following pdf contains slides to the talk I gave at the AMS Sectional in Tallahassee in 2024.