The Big Dehn Surgery Graph and the link of 𝑆³

Hoffman, Neil; Walsh, Genevieve

doi:10.1090/bproc/20

The Big Dehn Surgery Graph and the link of $S^3$
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by Neil R. Hoffman and Genevieve S. Walsh HTML | PDF

Proc. Amer. Math. Soc. Ser. B 2 (2015), 17-34

Abstract:

In a talk at the Cornell Topology Festival in 2004, W. Thurston discussed a graph which we call “The Big Dehn Surgery Graph”, $\mathcal {B}$. Here we explore this graph, particularly the link of $S^3$, and prove facts about the geometry and topology of $\mathcal {B}$. We also investigate some interesting subgraphs and pose what we believe are important questions about $\mathcal {B}$.

References

Matthias Aschenbrenner and Stefan Friedl, 3-manifold groups are virtually residually $p$, Mem. Amer. Math. Soc. 225 (2013), no. 1058, viii+100. MR 3100378, DOI 10.1090/S0065-9266-2013-00682-X
David Auckly, An irreducible homology sphere which is not Dehn surgery on any knot, preprint available at http://www.math.ksu.edu/~dav/.
David Auckly, Surgery numbers of $3$-manifolds: a hyperbolic example, Geometric topology (Athens, GA, 1993) AMS/IP Stud. Adv. Math., vol. 2, Amer. Math. Soc., Providence, RI, 1997, pp. 21–34. MR 1470719, DOI 10.1090/amsip/002.1/02
Steven Boyer, Dehn surgery on knots, Handbook of geometric topology, North-Holland, Amsterdam, 2002, pp. 165–218. MR 1886670
Steven Boyer and Daniel Lines, Surgery formulae for Casson’s invariant and extensions to homology lens spaces, J. Reine Angew. Math. 405 (1990), 181–220. MR 1041002
Marc Culler and Nathan M. Dunfield, SnapPy, a computer program for studying the geometry and topology of 3-manifolds. available at http://snappy.computop.org.
Marc Culler, C. McA. Gordon, J. Luecke, and Peter B. Shalen, Dehn surgery on knots, Ann. of Math. (2) 125 (1987), no. 2, 237–300. MR 881270, DOI 10.2307/1971311
Margaret I. Doig, Finite knot surgeries and Heegaard Floer homology, Algebr. Geom. Topol. 15 (2015), no. 2, 667–690. MR 3342672, DOI 10.2140/agt.2015.15.667
William D. Dunbar, Classification of solvorbifolds in dimension three. I, Braids (Santa Cruz, CA, 1986) Contemp. Math., vol. 78, Amer. Math. Soc., Providence, RI, 1988, pp. 207–216. MR 975080, DOI 10.1090/conm/078/975080
C. McA. Gordon, Dehn surgery and satellite knots, Trans. Amer. Math. Soc. 275 (1983), no. 2, 687–708. MR 682725, DOI 10.1090/S0002-9947-1983-0682725-0
C. McA. Gordon and J. Luecke, Knots are determined by their complements, Bull. Amer. Math. Soc. (N.S.) 20 (1989), no. 1, 83–87. MR 972070, DOI 10.1090/S0273-0979-1989-15706-6
C. McA. Gordon and J. Luecke, Toroidal and boundary-reducing Dehn fillings, Topology Appl. 93 (1999), no. 1, 77–90. MR 1684214, DOI 10.1016/S0166-8641(97)00258-7
Joshua Evan Greene and Liam Watson, Turaev torsion, definite 4-manifolds, and quasi-alternating knots, Bull. Lond. Math. Soc. 45 (2013), no. 5, 962–972. MR 3104988, DOI 10.1112/blms/bds096
Michael Gromov, Hyperbolic manifolds (according to Thurston and Jørgensen), Bourbaki Seminar, Vol. 1979/80, Lecture Notes in Math., vol. 842, Springer, Berlin-New York, 1981, pp. 40–53. MR 636516
Alan E. Hatcher, Notes on basic 3-manifold topology, available at http://www.math.cornell.edu/~hatcher/3M/3Mdownloads.html, 2000.
Damian Heard, Orb, available at www.ms.unimelb.edu.au/~snap/orb.html, 2005.
James Howie, A proof of the Scott-Wiegold conjecture on free products of cyclic groups, J. Pure Appl. Algebra 173 (2002), no. 2, 167–176. MR 1915093, DOI 10.1016/S0022-4049(02)00042-7
Kazuhiro Ichihara and Toshio Saito, Surgical distance between lens spaces, Tokyo J. Math. 34 (2011), no. 1, 153–164. MR 2866640, DOI 10.3836/tjm/1313074448
András Juhász, Holomorphic discs and sutured manifolds, Algebr. Geom. Topol. 6 (2006), 1429–1457. MR 2253454, DOI 10.2140/agt.2006.6.1429
P. Kutzko, On groups of finite weight, Proc. Amer. Math. Soc. 55 (1976), no. 2, 279–280. MR 399272, DOI 10.1090/S0002-9939-1976-0399272-8
W. B. R. Lickorish, A representation of orientable combinatorial $3$-manifolds, Ann. of Math. (2) 76 (1962), 531–540. MR 151948, DOI 10.2307/1970373
Marco Marengon, On $d$-invariants and generalised Kanenobu knots, arXiv preprint arXiv:1412.3433, 2014.
José María Montesinos, Classical tessellations and three-manifolds, Universitext, Springer-Verlag, Berlin, 1987. MR 915761, DOI 10.1007/978-3-642-61572-6
Louise Moser, Elementary surgery along a torus knot, Pacific J. Math. 38 (1971), 737–745. MR 383406, DOI 10.2140/pjm.1971.38.737
David Mullins, The generalized Casson invariant for $2$-fold branched covers of $S^3$ and the Jones polynomial, Topology 32 (1993), no. 2, 419–438. MR 1217078, DOI 10.1016/0040-9383(93)90029-U
Robert Myers, Excellent $1$-manifolds in compact $3$-manifolds, Topology Appl. 49 (1993), no. 2, 115–127. MR 1206219, DOI 10.1016/0166-8641(93)90038-F
Yi Ni, Link Floer homology detects the Thurston norm, Geom. Topol. 13 (2009), no. 5, 2991–3019. MR 2546619, DOI 10.2140/gt.2009.13.2991
Peter Ozsváth and Zoltán Szabó, Absolutely graded Floer homologies and intersection forms for four-manifolds with boundary, Adv. Math. 173 (2003), no. 2, 179–261. MR 1957829, DOI 10.1016/S0001-8708(02)00030-0
Peter Ozsváth and Zoltán Szabó, Holomorphic disks and genus bounds, Geom. Topol. 8 (2004), 311–334. MR 2023281, DOI 10.2140/gt.2004.8.311
Peter S. Ozsváth and Zoltán Szabó, Knot Floer homology and rational surgeries, Algebr. Geom. Topol. 11 (2011), no. 1, 1–68. MR 2764036, DOI 10.2140/agt.2011.11.1
Martin Scharlemann, Crossing changes, Chaos Solitons Fractals 9 (1998), no. 4-5, 693–704. Knot theory and its applications. MR 1628751, DOI 10.1016/S0960-0779(97)00105-7
Peter Scott, The geometries of $3$-manifolds, Bull. London Math. Soc. 15 (1983), no. 5, 401–487. MR 705527, DOI 10.1112/blms/15.5.401
William Thurston, The geometry and topology of 3-manifolds. Princeton University, Mimeographed lecture notes, 1977.
William P. Thurston, Hyperbolic structures on $3$-manifolds. I. Deformation of acylindrical manifolds, Ann. of Math. (2) 124 (1986), no. 2, 203–246. MR 855294, DOI 10.2307/1971277
Andrew H. Wallace, Modifications and cobounding manifolds, Canadian J. Math. 12 (1960), 503–528. MR 125588, DOI 10.4153/CJM-1960-045-7