Genus one cobordisms between torus knots

Abstract

We determine the pairs of torus knots that have a genus one cobordism between them, with one notable exception. This is done by combining obstructions using ${\nu }^{+}$ from the Heegaard Floer knot complex and explicit constructions of cobordisms. As an application, we determine the pairs of torus knots related by a single crossing change. Also, we determine the pairs of Thurston-Bennequin number maximizing Legendrian torus knots that have a genus one exact Lagrangian cobordism, with one exception.

20 pages, 12 figures. V3: Minor corrections and implementation of referee's recommendations. This version has been accepted for publication by IMRN