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A distance for circular Heegaard splittings

Kevin Lamb

http://orcid.org/0000-0002-1265-5050

Department of Mathematics, University of the Pacific, CA, USA

E-mail Address: klamb@pacific.edu

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and

Patrick Weed

Department of Mathematics, University of California, Davis, CA, USA

E-mail Address: psweed@math.ucdavis.edu

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https://doi.org/10.1142/S0218216519500597Cited by:1 (Source: Crossref)

Abstract

For a knot $K \subset S^{3}$ , its exterior $E (K) = S^{3} ∖ η (K)$ has a singular foliation by Seifert surfaces of $K$ derived from a circle-valued Morse function $f : E (K) \to S^{1}$ . When $f$ is self-indexing and has no critical points of index 0 or 3, the regular levels that separate the index-1 and index-2 critical points decompose $E (K)$ into a pair of compression bodies. We call such a decomposition a circular Heegaard splitting of $E (K)$ . We define the notion of circular distance (similar to Hempel distance) for this class of Heegaard splitting and show that it can be bounded under certain circumstances. Specifically, if the circular distance of a circular Heegaard splitting is too large: (1) $E (K)$ cannot contain low-genus incompressible surfaces, and (2) a minimal-genus Seifert surface for $K$ is unique up to isotopy.

Keywords:

AMSC: 57-XX

References

1. D. Bachman and S. Schleimer, Distance and bridge position, Pacific J. Math. 219(2) (2005) 221–235. Crossref, Web of Science, Google Scholar
2. J. Birman, The topology of 3-manifolds, Heegaard distance and the mapping class group of a 2-manifold, in Problems on Mapping Class Groups and Related Topics, Proceedings of Symposia in Pure Mathematics, Vol. 74 (American Mathematical Society, Providence, RI, 2006), pp. 133–149. Crossref, Google Scholar
3. F. Bonahon and J. P. Otal, Scindements de Heegaard des espaces Lenticulaires, Ann. Sci. Ecole Norm. Sup. 16 (1983) 451–466. Crossref, Web of Science, Google Scholar
4. A. Casson and C. McA. Gordon, Reducibility of Heegaard splittings, Topol. Appl. 27 (1987) 275–283. Crossref, Web of Science, Google Scholar
5. T. Evans, High distance Heegaard splittings of 3-manifolds, Topol. Appl. 15 (2006) 2631–2647. Crossref, Web of Science, Google Scholar
6. K. Hartshorn, Heegaard splittings of Haken manifolds have bounded distance, Pacific J. Math. 204(1) (2002) 61–75. Crossref, Web of Science, Google Scholar
7. W. J. Harvey, Geometric structure of surface mapping class groups, in Homological Group Theory, London Mathematical Society Lecture Note Series, Vol. 36 (Cambridge University Press, Cambridge, 1979), pp. 255–269. Crossref, Google Scholar
8. J. Hempel, 3-manifolds as viewed from the curve complex, Topology 40 (2001) 631–567. Crossref, Web of Science, Google Scholar
9. W. H. Jaco, Lectures on three-manifold topology, CBMS Regional Conference Series in Mathematics, Vol. 43 (American Mathematical Society, Providence, RI, 1980), xii+251 pp. Crossref, Google Scholar
10. J. Johnson, Heegaard splittings and open books, preprint (2011), arXiv:1110.2142v1 [math.GT]. Google Scholar
11. T. Kobayashi, Heights of simple loops and pseudo-Anosov homeomorphisms, in Braids American Mathematical Society, Providence, RI, 1988), pp. 327–338. Google Scholar
12. F. Manjarrez-Gutiérrez, Circular thin position for knots in $S^{3}$ , Algebr. Geom. Topol. 9 (2009) 429–454. Crossref, Web of Science, Google Scholar
13. F. Manjarrez-Gutiérrez, Additivity of handle number and Morse-Novikov number of a-small knots, Topol. Appl. 160 (2013) 117–125. Crossref, Web of Science, Google Scholar
14. J. Milnor, Lectures on the h-Cobordism Theorem (Princeton University Press, Princeton, NJ, 1963). Google Scholar
15. E. E. Moise, Affine structures in 3-manifolds: V. The triangulation theorem and Hauptermutung, Ann. Math. 56(1) 96–114 (1952). Crossref, Web of Science, Google Scholar
16. D. Rolfsen, Knots and Links (Publish or Perish, Berkeley, CA, 1976). Google Scholar
17. M. Scharlemann and A. Thompson, Finding disjoint Seifert surfaces, Bull. London Math. Soc. 20 (1988) 61–64. Crossref, Web of Science, Google Scholar
18. M. Scharlemann and A. Thompson, Thin position for 3-manifolds, Contemp. Math. 164 (1994) 231–238. Crossref, Google Scholar
19. M. Scharlemann and M. Tomova, Alternate Heegaard genus bounds distance, Geom. Topol. 10(1) (2006) 593–617. Crossref, Web of Science, Google Scholar
20. A. Thompson, The disjoint curve property and genus 2 manifolds, Topol. Appl. 93(3) (1999) 273–279. Crossref, Web of Science, Google Scholar