Comparing 2-handle additions to a genus 2 boundary component
HTML articles powered by AMS MathViewer
- by Scott A. Taylor PDF
- Trans. Amer. Math. Soc. 366 (2014), 3747-3769 Request permission
Abstract:
We prove that knots obtained by attaching a band to a split link satisfy the cabling conjecture. We also give new proofs that unknotting number one knots are prime and that genus is superadditive under a band sum. Additionally, we prove a collection of results comparing two 2-handle additions to a genus 2 boundary component of a compact, orientable 3-manifold. These results give a near complete solution to a conjecture of Scharlemann and provide evidence for a conjecture of Scharlemann and Wu. The proofs make use of a new theorem concerning the effects of attaching a 2-handle to a suture in the boundary of a sutured manifold.References
- Mario Eudave Muñoz, Primeness and sums of tangles, Trans. Amer. Math. Soc. 306 (1988), no. 2, 773–790. MR 933317, DOI 10.1090/S0002-9947-1988-0933317-1
- Mario Eudave Muñoz, On nonsimple $3$-manifolds and $2$-handle addition, Topology Appl. 55 (1994), no. 2, 131–152. MR 1256216, DOI 10.1016/0166-8641(94)90114-7
- David Gabai, Foliations and the topology of $3$-manifolds, J. Differential Geom. 18 (1983), no. 3, 445–503. MR 723813
- David Gabai, Genus is superadditive under band connected sum, Topology 26 (1987), no. 2, 209–210. MR 895573, DOI 10.1016/0040-9383(87)90061-9
- David Gabai, Foliations and the topology of $3$-manifolds. II, J. Differential Geom. 26 (1987), no. 3, 461–478. MR 910017
- David Gabai, Foliations and the topology of $3$-manifolds. III, J. Differential Geom. 26 (1987), no. 3, 479–536. MR 910018
- Francisco González-Acuña and Hamish Short, Knot surgery and primeness, Math. Proc. Cambridge Philos. Soc. 99 (1986), no. 1, 89–102. MR 809502, DOI 10.1017/S0305004100063969
- C. McA. Gordon and J. Luecke, Only integral Dehn surgeries can yield reducible manifolds, Math. Proc. Cambridge Philos. Soc. 102 (1987), no. 1, 97–101. MR 886439, DOI 10.1017/S0305004100067086
- Chuichiro Hayashi, Tangles and tubing operations, Topology Appl. 92 (1999), no. 3, 191–199. MR 1674809, DOI 10.1016/S0166-8641(97)00250-2
- William Jaco, Adding a $2$-handle to a $3$-manifold: an application to property $R$, Proc. Amer. Math. Soc. 92 (1984), no. 2, 288–292. MR 754723, DOI 10.1090/S0002-9939-1984-0754723-6
- Yan Nan Li, Rui Feng Qiu, and Ming Xing Zhang, Boundary reducible handle additions on simple 3-manifolds, Acta Math. Sin. (Engl. Ser.) 25 (2009), no. 2, 235–244. MR 2481491, DOI 10.1007/s10114-008-7119-y
- Martin G. Scharlemann, Unknotting number one knots are prime, Invent. Math. 82 (1985), no. 1, 37–55. MR 808108, DOI 10.1007/BF01394778
- Martin Scharlemann, The Thurston norm and $2$-handle addition, Proc. Amer. Math. Soc. 100 (1987), no. 2, 362–366. MR 884480, DOI 10.1090/S0002-9939-1987-0884480-7
- Martin Scharlemann, Sutured manifolds and generalized Thurston norms, J. Differential Geom. 29 (1989), no. 3, 557–614. MR 992331
- Martin Scharlemann, Producing reducible $3$-manifolds by surgery on a knot, Topology 29 (1990), no. 4, 481–500. MR 1071370, DOI 10.1016/0040-9383(90)90017-E
- Martin Scharlemann, Refilling meridians in a genus 2 handlebody complement, The Zieschang Gedenkschrift, Geom. Topol. Monogr., vol. 14, Geom. Topol. Publ., Coventry, 2008, pp. 451–475. MR 2484713, DOI 10.2140/gtm.2008.14.451
- Martin Scharlemann and Abigail Thompson, Unknotting number, genus, and companion tori, Math. Ann. 280 (1988), no. 2, 191–205. MR 929535, DOI 10.1007/BF01456051
- Martin Scharlemann and Ying Qing Wu, Hyperbolic manifolds and degenerating handle additions, J. Austral. Math. Soc. Ser. A 55 (1993), no. 1, 72–89. MR 1231695
- Scott Allen Taylor, Boring split links and unknots, ProQuest LLC, Ann Arbor, MI, 2008. Thesis (Ph.D.)–University of California, Santa Barbara. MR 2712342
- Scott A. Taylor, Boring split links, Pacific J. Math. 241 (2009), no. 1, 127–167. MR 2485461, DOI 10.2140/pjm.2009.241.127
- Scott A. Taylor, Band-taut sutured manifolds, Algebr. Geom. Topol. 14 (2014), no. 1, 157–215. MR 3158757, DOI 10.2140/agt.2014.14.157
- Mingxing Zhang, Ruifeng Qiu, and Yannan Li, The distance between two separating, reducing slopes is at most 4, Math. Z. 257 (2007), no. 4, 799–810. MR 2342554, DOI 10.1007/s00209-007-0147-y
Additional Information
- Scott A. Taylor
- Affiliation: Department of Mathematics and Statistics, Colby College, Waterville, Maine 04901
- Email: sataylor@colby.edu
- Received by editor(s): November 3, 2011
- Received by editor(s) in revised form: October 12, 2012
- Published electronically: March 4, 2014
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 366 (2014), 3747-3769
- MSC (2010): Primary 57M50
- DOI: https://doi.org/10.1090/S0002-9947-2014-06253-3
- MathSciNet review: 3192616