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 sub-set 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.
My 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:
- upcoming: Title TBD, Oct 6 2024, Vanderbilt Topology Seminar
- upcoming: 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>
- “Possible fundamental groups for genus-2 Lefschetz fibrations,” Oct 2023, Georgia Tech Student Topology Seminar
- “An introduction to 4-manifolds,” Sep 2023, Georgia Tech Student Topology Seminar
- “An exotic 4-manifold,” Feb 2022, Georgia Tech Student Topology Seminar
- “A Lefschetz fibration construction with arbitrary finitely presented fundamental group,” July 2023, Tech Topology Summer School <link>
- “An introduction to 4-manifolds,” Sep 2022, Georgia Tech Student Topology Seminar
- “Constructing Exotic 4-manifolds via cork twists, Gluck twists, and log transforms,” Dec 2021, Georgia Tech Student Topology Seminar
The following pdf contains slides to the talk I gave at the AMS Sectional in Tallahassee.