Semigroups of L-space knots and nonalgebraic iterated torus knots

TL;DR

This paper constructs an infinite family of iterated torus knots that admit L-space surgeries but are not algebraic, introducing a new invariant called the formal semigroup to distinguish them.

Contribution

It provides the first infinite family of nonalgebraic iterated torus knots with L-space surgeries and introduces the formal semigroup invariant.

Findings

Identifies an infinite family of nonalgebraic L-space knots.

Introduces the formal semigroup as a new invariant.

Uses the Upsilon function to distinguish knots.

Abstract

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 Ozsvath-Stipsicz-Szabo Upsilon function, and also introduces a new invariant of L-space knots, the formal semigroup.