Abstract
Blair, Campisi, Taylor, and Tomova introduced a non-negative integer-valued invariant \({\mathcal {L}}(S)\) of a smooth surface S in the 4-sphere. In this paper, we extend previous work done by the authors with Scott Taylor to compute the invariant \({\mathcal {L}}(S)\) of a knotted surface in 4-space. We further explore the combinatorics of pants decompositions to give sharp bounds for the \({\mathcal {L}}\)-invariant of large families of bridge trisections. As an application, we show that surfaces with \({\mathcal {L}}(S)\le 2\) must be unknotted.
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References
Aranda, R., Pongtanapaisan, P., Taylor, S.A., Zhang, S.: Bounding the kirby-thompson invariant of spun knots. arXiv preprint arXiv:2112.02420 (2021)
Artin, E.: Zur isotopie zweidimensionaler flächen imr4. Abh. Math. Semin. Univ. Hambg. 4, 174–177 (1925)
Bachman, D., Schleimer, S.: Distance and bridge position. Pac. J. Math. 219(2), 221–235 (2005)
Blair, R., Campisi, M., Taylor, S.A., Tomova, M.: Kirby–Thompson distance for trisections of knotted surfaces. J. London Math. Soc. (2021)
Hatcher, A., Thurston, W.: Incompressible surfaces in \(2\)-bridge knot complements. Invent. Math. 79(2), 225–246 (1985). https://doi.org/10.1007/BF01388971
Hayashi, C., Shimokawa, K.: Heegaard splittings of the trivial knot. J. Knot Theory Ramifications 7(8), 1073–1085 (1998). https://doi.org/10.1142/S0218216598000589
Joseph, J., Meier, J., Miller, M., Zupan, A.: Bridge trisections and classical knotted surface theory (2021). https://doi.org/10.48550/ARXIV.2112.11557. arXiv:2112.11557
Joseph, J., Pongtanapaisan, P.: Bridge numbers and meridional ranks of knotted surfaces and welded knots. arXiv preprint arXiv:2111.09233 (2021)
Kirby, R., Thompson, A.: A new invariant of 4-manifolds. Proc. Natl. Acad. Sci. 115(43), 10857–10860 (2018)
Livingston, C.: Stably irreducible surfaces in s4. Pac. J. Math. 116(1), 77–84 (1985)
Meier, J.: Filling braided links with trisected surfaces. arXiv preprint arXiv:2010.11135 (2020)
Meier, J., Zupan, A.: Bridge trisections of knotted surfaces in \(S^4\). Trans. Am. Math. Soc. 369(10), 7343–7386 (2017)
Ogawa, M.: Trisections with kirby–thompson length 2. arXiv preprint arXiv:2112.10033 (2021)
Otal, J.P.: Présentations en ponts du nœud trivial. C. R. Acad. Sci. Paris Sér. I Math. 294(16), 553–556 (1982)
Yoshikawa, K.: An enumeration of surfaces in four-space. Osaka J. Math. 31(3), 497–522 (1994)
Zupan, A.: Bridge and pants complexities of knots. J. Lond. Math. Soc. 87(1), 43–68 (2013)
Acknowledgements
We are grateful to Scott Taylor for his constant advice. We thank Jeffrey Meier for helpful comments on the draft. PP acknowledges the Pacific Institute for the Mathematical Sciences for the support. Some of this work was done when RA visited Colby College in Spring 2022.
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Aranda, R., Pongtanapaisan, P. & Zhang, C. Bounds for Kirby–Thompson invariants of knotted surfaces. Geom Dedicata 217, 99 (2023). https://doi.org/10.1007/s10711-023-00835-6
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DOI: https://doi.org/10.1007/s10711-023-00835-6