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The four-genus of connected sums of torus knots

Published online by Cambridge University Press:  17 April 2017

CHARLES LIVINGSTON  and
CORNELIA A. VAN COTT
CHARLES LIVINGSTON
Affiliation:
Department of Mathematics, Indiana University, Bloomington, Indiana 47405, U.S.A. e-mail: [email protected]
CORNELIA A. VAN COTT
Affiliation:
Department of Mathematics and Statistics, University of San Francisco, San Francisco, CA 94117, U.S.A e-mail: [email protected]
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Abstract

We study the four-genus of linear combinations of torus knots: g4(aT(p, q) #-bT(p′, q′)). Fixing positive p, q, p′, and q′, our focus is on the behavior of the four-genus as a function of positive a and b. Three types of examples are presented: in the first, for all a and b the four-genus is completely determined by the Tristram–Levine signature function; for the second, the recently defined Upsilon function of Ozsváth–Stipsicz–Szabó determines the four-genus for all a and b; for the third, a surprising interplay between signatures and Upsilon appears.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 164 , Issue 3 , May 2018 , pp. 531 - 550
Copyright
Copyright © Cambridge Philosophical Society 2017 

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