Semigroups of $L$-space knots and nonalgebraic iterated torus knots

Algebraic knots are known to be iterated torus knots and to admit $L$-space surgeries. However, Hedden proved that there are iterated torus knots that admit $L$-space surgeries but are not algebraic. We present an infinite family of such examples, with the additional property that no nontrivial linear combination of knots in this family is concordant to a linear combination of algebraic knots. The proof uses the Ozsváth–Stipsicz–Szabó Upsilon function, and also introduces a new invariant of $L$-space knots, the formal semigroup.

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Received 6 August 2016

Accepted 30 April 2017

Published 4 June 2018