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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.

References

Barbensi, A. and Celoria, D., ‘GridPyM: a Python module to handle grid diagrams’, J. Softw. Algebra Geom. 14(1) (2024), 31–39.CrossRefGoogle Scholar
Burton, B. A., ‘The next 350 million knots’, in: 36th International Symposium on Computational Geometry, SoCG 2020, Zürich, Switzerland, Leibniz International Proceedings in Informatics, 164 (eds. Cabello, S. and Chen, D. Z.) (Schloss Dagstuhl, Leibniz Zentrum für Informatik, 2020), Article no. 17, 25 pages.Google Scholar
Culler, M., Dunfield, N. M., Goerner, M. and Weeks, J. R., ‘SnapPy, a computer program for studying the geometry and topology of 3-manifolds’. Available online at http://snappy.computop.org.Google Scholar
Livingston, C., ‘Computations of the Ozsváth–Szabó knot concordance invariant’, Geom. Topol. 8(2) (2004), 735–742.CrossRefGoogle Scholar
Manolescu, C., Ozsváth, P., Szabó, Z. and Thurston, D., ‘On combinatorial link Floer homology’, Geom. Topol. 11 (2007), 2339–2412.CrossRefGoogle Scholar
Mikels, P. and Guerra, O., Simplifying knots through algorithmic procedures. Unpublished undergraduate thesis, 2017.Google Scholar
Ozsváth, P. and Szabó, Z., ‘Knot Floer homology and the four-ball genus’, Geom. Topol. 7 (2003), 615–639.CrossRefGoogle Scholar
Ozsváth, P. and Szabó, Z., ‘Knot Floer homology calculator’. Available online at https://web.math.princeton.edu/~szabo/HFKcalc.html (accessed 20 November 2024).Google Scholar
Sarkar, S., ‘Grid diagrams and the Ozsváth–Szabó tau-invariant’, Math. Res. Lett. 18(6) (2011), 1239–1257.CrossRefGoogle Scholar
The Knot Atlas: 36 Torus Knots. Available online at http://katlas.org/wiki/36_Torus_Knots (accessed 11 July 2019).Google Scholar
Thurston, W. P., ‘Three-dimensional manifolds, Kleinian groups and hyperbolic geometry’, Bull. Amer. Math. Soc. (N.S.) 6(3) (1982), 357–381.CrossRefGoogle Scholar