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Tunnel number one knots, $ m$-small knots and the Morimoto conjecture


Authors: Guoqiu Yang, Xunbo Yin and Fengchun Lei
Journal: Proc. Amer. Math. Soc. 141 (2013), 4391-4399
MSC (2010): Primary 57M99
DOI: https://doi.org/10.1090/S0002-9939-2013-11700-4
Published electronically: August 16, 2013
MathSciNet review: 3105881
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Abstract | References | Similar Articles | Additional Information

Abstract: In the present paper, we show that the Morimoto Conjecture on the super additivity of the tunnel numbers of knots in $ S^3$ is true for knots $ K_1,K_2$ in $ S^3$ in which each $ K_i$ is either a tunnel number one or $ m$-small, $ i=1,2$. This extends two known results by Morimoto.


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  • [1] A. J. Casson and C. McA. Gordon, Reducing Heegaard splittings, Topology Appl. 27 (1987), no. 3, 275-283. MR 918537 (89c:57020), https://doi.org/10.1016/0166-8641(87)90092-7
  • [2] Tsuyoshi Kobayashi, A construction of arbitrarily high degeneration of tunnel numbers of knots under connected sum, J. Knot Theory Ramifications 3 (1994), no. 2, 179-186. MR 1279920 (95g:57011), https://doi.org/10.1142/S0218216594000137
  • [3] Tsuyoshi Kobayashi and Yo'av Rieck, Heegaard genus of the connected sum of $ m$-small knots, Comm. Anal. Geom. 14 (2006), no. 5, 1037-1077. MR 2287154 (2007i:57018)
  • [4] Tsuyoshi Kobayashi and Yo'av Rieck, Knot exteriors with additive Heegaard genus and Morimoto's conjecture, Algebr. Geom. Topol. 8 (2008), no. 2, 953-969. MR 2443104 (2009m:57011), https://doi.org/10.2140/agt.2008.8.953
  • [5] Tsuyoshi Kobayashi and Yo'av Rieck, Knots with $ g(E(K))=2$ and $ g(E(K\,\char93 \,K\,\char93 \,K))=6$ and Morimoto's conjecture, Topology Appl. 156 (2009), no. 6, 1114-1117. MR 2493371 (2010b:57007), https://doi.org/10.1016/j.topol.2008.10.003
  • [6] Martin Lustig and Yoav Moriah, Generalized Montesinos knots, tunnels and $ \mathcal {N}$-torsion, Math. Ann. 295 (1993), no. 1, 167-189. MR 1198847 (94b:57011), https://doi.org/10.1007/BF01444882
  • [7] Martin Lustig and Yoav Moriah, Closed incompressible surfaces in complements of wide knots and links, Topology Appl. 92 (1999), no. 1, 1-13. MR 1670164 (2000b:57009), https://doi.org/10.1016/S0166-8641(97)00232-0
  • [8] Yoav Moriah, Heegaard splittings of knot exteriors, Workshop on Heegaard Splittings, Geom. Topol. Monogr., vol. 12, Geom. Topol. Publ., Coventry, 2007, pp. 191-232. MR 2408247 (2009i:57015), https://doi.org/10.2140/gtm.2007.12.191
  • [9] Yoav Moriah and Hyam Rubinstein, Heegaard structures of negatively curved $ 3$-manifolds, Comm. Anal. Geom. 5 (1997), no. 3, 375-412. MR 1487722 (98j:57029)
  • [10] Kanji Morimoto, Tunnel number, connected sum and meridional essential surfaces, Topology 39 (2000), no. 3, 469-485. MR 1746903 (2001a:57015), https://doi.org/10.1016/S0040-9383(98)00070-6
  • [11] Kanji Morimoto, On the additivity of tunnel number of knots, Topology Appl. 53 (1993), no. 1, 37-66. MR 1243869 (94j:57011), https://doi.org/10.1016/0166-8641(93)90099-Y
  • [12] Kanji Morimoto, On the super additivity of tunnel number of knots, Math. Ann. 317 (2000), no. 3, 489-508. MR 1776114 (2001g:57016), https://doi.org/10.1007/PL00004411
  • [13] Kanji Morimoto, There are knots whose tunnel numbers go down under connected sum, Proc. Amer. Math. Soc. 123 (1995), no. 11, 3527-3532. MR 1317043 (96a:57022), https://doi.org/10.2307/2161103
  • [14] Kanji Morimoto and Jennifer Schultens, Tunnel numbers of small knots do not go down under connected sum, Proc. Amer. Math. Soc. 128 (2000), no. 1, 269-278. MR 1641065 (2000c:57014), https://doi.org/10.1090/S0002-9939-99-05160-6
  • [15] Kanji Morimoto, Makoto Sakuma, and Yoshiyuki Yokota, Examples of tunnel number one knots which have the property $ \lq\lq 1+1=3''$, Math. Proc. Cambridge Philos. Soc. 119 (1996), no. 1, 113-118. MR 1356163 (96i:57007), https://doi.org/10.1017/S0305004100074028
  • [16] Jennifer Schultens, The classification of Heegaard splittings for (compact orientable surface) $ \,\times \, S^1$, Proc. London Math. Soc. (3) 67 (1993), no. 2, 425-448. MR 1226608 (94d:57043), https://doi.org/10.1112/plms/s3-67.2.425
  • [17] Jennifer Schultens, Additivity of tunnel number for small knots, Comment. Math. Helv. 75 (2000), no. 3, 353-367. MR 1793793 (2001i:57012), https://doi.org/10.1007/s000140050131
  • [18] Martin Scharlemann and Abigail Thompson, Thin position for $ 3$-manifolds, Geometric topology (Haifa, 1992) Contemp. Math., vol. 164, Amer. Math. Soc., Providence, RI, 1994, pp. 231-238. MR 1282766 (95e:57032), https://doi.org/10.1090/conm/164/01596
  • [19] Martin Scharlemann, Local detection of strongly irreducible Heegaard splittings, Topology Appl. 90 (1998), no. 1-3, 135-147. MR 1648310 (99h:57040), https://doi.org/10.1016/S0166-8641(97)00184-3
  • [20] Martin Scharlemann and Jennifer Schultens, The tunnel number of the sum of $ n$ knots is at least $ n$, Topology 38 (1999), no. 2, 265-270. MR 1660345 (2000b:57013), https://doi.org/10.1016/S0040-9383(98)00002-0

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Additional Information

Guoqiu Yang
Affiliation: School of Astronautics and Department of Mathematics, Harbin Institute of Technology, Harbin 150001, People’s Republic of China
Email: gqyang@hit.edu.cn

Xunbo Yin
Affiliation: Department of Mathematics, Harbin Institute of Technology, Harbin 150001, People’s Republic of China
Email: jlxbyin@hit.edu.cn

Fengchun Lei
Affiliation: School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, People’s Republic of China
Email: ffcclei@yahoo.com.cn

DOI: https://doi.org/10.1090/S0002-9939-2013-11700-4
Keywords: Tunnel number, $m$-small, Morimoto Conjecture
Received by editor(s): July 15, 2011
Received by editor(s) in revised form: November 8, 2011, December 20, 2011, January 18, 2012, and February 8, 2012
Published electronically: August 16, 2013
Additional Notes: The first author was supported in part by two grants of NSFC (No. 11001065 and No. 11071106) and by two grants of HITQNJS (No. 2009.029 and No. 20100471066)
The second author was supported in part by a grant of NSFC (No. 11001065)
The third author was supported in part by a key grant of NSFC (No. 10931005)
Communicated by: Daniel Ruberman
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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