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DIAGONAL KNOTS AND THE TAU INVARIANT

Published online by Cambridge University Press:  06 February 2025

JACKSON ARNDT ,
MALIA JANSEN ,
PAYTON MCBURNEY  and
KATHERINE VANCE Open the ORCID record for KATHERINE VANCE [Opens in a new window]
JACKSON ARNDT
Affiliation:
Department of Mathematics and Computer Science, Simpson College, Indianola, IA 50125, USA
MALIA JANSEN
Affiliation:
Department of Mathematics and Computer Science, Simpson College, Indianola, IA 50125, USA
PAYTON MCBURNEY
Affiliation:
Department of Mathematics and Computer Science, Simpson College, Indianola, IA 50125, USA
KATHERINE VANCE*
Affiliation:
Department of Mathematics and Computer Science, Simpson College, Indianola, IA 50125, USA
*
e-mail: [email protected]
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Abstract

In 2003, Ozsváth, Szabó and Rasmussen introduced the $\tau $ invariant for knots, and in 2011, Sarkar [‘Grid diagrams and the Ozsváth–Szabó tau-invariant’, Math. Res. Lett. 18(6) (2011), 1239–1257] published a computational shortcut for the $\tau $ invariant of knots that can be represented by diagonal grid diagrams. Previously, the only knots known to have diagonal grid diagram representations were torus knots. We prove that all such knots are positive knots and we produce an example of a knot with a diagonal grid diagram representation which is not a torus knot.

Keywords

knot theorygrid diagramstau invariant
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , First View , pp. 1 - 7
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

This research was funded by Dr. Albert H. and Greta A. Bryan through the 2017 Bryan Summer Research Program at Simpson College.

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