Positivities of Knots and Links and the Defect of Bennequin Inequality

Abstract

We discuss relations among various positivities of knots and links, such as strong quasipositivity and quasipositivity. We give several pieces of supporting evidence for conjectural statements concerning these positivities and the defect of Bennequin inequality. Finally, we determine strong quasipositivity and quasipositivity for knots up to 12 crossings (with two exceptions for quasipositivity).

Keywords:

• Bennequin inequality
• defect
• strongly quasipositive
• quasipositive
• homogeneous link

Acknowledgments

The authors would like to thank Jeremy Van Horn-Morris for sharing the list of knots whose strongly quasipositivity was previously unknown, Camila Ramilez for making an Excel file, and Inanc Baykur, Daryl Cooper, John Etnyre, Charlie Frohman, and Matthew Hedden for useful conversations. They also thank the anonymous referee for useful comments and suggestions.

Additional information

Funding

JH was partially supported by The University of Iowa’s Ballard and Seashore Dissertation Fellowship. TI was partially supported by JSPS Grant-in-Aid for Young Scientists (B) 15K17540. KK was partially supported by NSF grant DMS-1206770 and Simons Foundation Collaboration Grants for Mathematicians.