 

Samantha Allen
Nonorientable surfaces bounded by knots: a geography problem
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Published: 
September 14, 2023. 
Keywords: 
Knot, Genus, Nonorientable, Euler number. 
Subject [2020]: 
57K10. 


Abstract
The nonorientable 4genus is an invariant of knots which has been studied by many authors, including Gilmer and Livingston, Batson, and
Ozsváth, Stipsicz, and Szabó. Given a nonorientable surface
F in B^{4} with ∂ F = K in S^{3} a knot, an analysis
of the existing methods for bounding and computing the nonorientable 4genus
reveals relationships between the first Betti number β_{1} of
F and the normal Euler class e of F. This relationship yields a geography
problem: given a knot K, what is the set of realizable pairs
(e(F), β_{1}(F)) where F in B^{4} is a nonorientable
surface bounded by K? We explore this problem for families of torus knots.
In addition, we use the OzsváthSzabó dinvariant of twofold
branched covers to give finer information on the geography problem. We
present an infinite family of knots where this information provides an
improvement upon the bound given by Ozsváth, Stipsicz, and
Szabó using the Upsilon invariant.


Acknowledgements
Thanks are due to Charles Livingston for guidance
and careful reading of many early versions of this paper. In addition,
Ina Petkova provided many helpful comments and suggestions for improving
the exposition.


Author information
Samantha Allen
Department of Mathematics and Computer Science
Duquesne University
Pittsburgh, PA 15217, USA
allens6@duq.edu

