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Kirby-Thompson distance for trisections of knotted surfaces
Journal article

Kirby-Thompson distance for trisections of knotted surfaces

Ryan Blair, Marion Campisi, Scott A Taylor and Maggy Tomova
Journal of the London Mathematical Society, Vol.105(2), pp.765-793
03/01/2022
DOI: 10.1112/jlms.12513
url
https://arxiv.org/pdf/2002.03991View
Open Access

Abstract

We adapt work of Kirby-Thompson and Zupan to define an integer invariant L(T)$\mathcal {L}(\mathcal {T})$ of a bridge trisection T$\mathcal {T}$ of a smooth surface S$S$ in S4$S<^>4$ or B4$B<^>4$. We show that when L(T)=0$\mathcal {L}(\mathcal {T})=0$, then the surface S$S$ is unknotted. We also show that for a trisection T$\mathcal {T}$ of an irreducible surface, bridge number produces a lower bound for L(T)$\mathcal {L}(\mathcal {T})$. Consequently L$\mathcal {L}$ can be arbitrarily large.
Mathematics Physical Sciences Science & Technology

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