Abstract
The algebraic genus of a knot is an invariant that arises when one considers upper bounds for the topological slice genus coming from Freedman’s theorem that Alexander polynomial one knots are topologically slice. This paper develops null-homologous twisting operations as a tool for studying the algebraic genus and, consequently, for bounding the topological slice genus above. As applications we give new upper bounds on the algebraic genera of torus knots and satellite knots.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Baader, S., Feller, P., Lewark, L., Liechti, L.: On the topological 4-genus of torus knots. Trans. Amer. Math. Soc. 370(4), 2639–2656 (2018)
Baader, S., Lewark, L.: The stable 4-genus of alternating knots. Asian J. Math. 21(6), 1183–1190 (2017)
Borodzik, M., Friedl, S.: On the algebraic unknotting number. Trans. London Math. Soc. 1(1), 57–84 (2014)
Borodzik, M., Friedl, S.: The unknotting number and classical invariants, I. Algebr. Geom. Topol. 15(1), 85–135 (2015)
Feller, P.: The degree of the Alexander polynomial is an upper bound for the topological slice genus. Geom. Topol. 20(3), 1763–1771 (2016)
Feller, P., Lewark, L.: On classical upper bounds for slice genera. Selecta Math. (N.S.) 24(5), 4885–4916 (2018)
Feller, P., Lewark, L.: Balanced algebraic unknotting, linking forms, and surfaces in three- and four-space. arXiv:1905.08305 (2019)
Feller, P., McCoy, D.: On 2-bridge knots with differing smooth and topological slice genera. Proc. Amer. Math. Soc. 144(12), 5435–5442 (2016)
Feller, P., Miller, A.N., Pinzon-Caicedo, J.: A note on the topological slice genus of satellite knots. arXiv:1908.03760 (2019)
Freedman, M.H.: The topology of four-dimensional manifolds. J. Differential Geom. 17(3), 357–453 (1982)
Lewark, L., McCoy, D.: On calculating the slice genera of 11- and 12-crossing knots. Exp. Math. 28(1), 81–94 (2019)
Lickorish, W.R.: An Introduction to Knot Theory. Springer (1997)
Liechti, L.: Positive braid knots of maximal topological 4-genus. Math. Proc. Cambridge Philos. Soc. 161(3), 559–568 (2016)
Livingston, C.: Null-homologous unknottings. arXiv:1902.05405 (2019)
Rudolph, L.: Some topologically locally-flat surfaces in the complex projective plane. Comment. Math. Helv. 59(4), 592–599 (1984)
Taylor, L.R.: On the genera of knots. In: Topology of low-dimensional manifolds (Proc. Second Sussex Conf., Chelwood Gate, 1977), Lecture Notes in Math., vol. 722, pp. 144–154. Springer, Berlin (1979)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
McCoy, D. (2021). Null-Homologous Twisting and the Algebraic Genus. In: de Gier, J., Praeger, C.E., Tao, T. (eds) 2019-20 MATRIX Annals. MATRIX Book Series, vol 4. Springer, Cham. https://doi.org/10.1007/978-3-030-62497-2_7
Download citation
DOI: https://doi.org/10.1007/978-3-030-62497-2_7
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-62496-5
Online ISBN: 978-3-030-62497-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)