## Algebraic & Geometric Topology

### Thin position for a connected sum of small knots

#### Abstract

We show that every thin position for a connected sum of small knots is obtained in an obvious way: place each summand in thin position so that no two summands intersect the same level surface, then connect the lowest minimum of each summand to the highest maximum of the adjacent summand below.

#### Article information

Source
Algebr. Geom. Topol., Volume 2, Number 1 (2002), 297-309.

Dates
Accepted: 6 March 2002
First available in Project Euclid: 21 December 2017

https://projecteuclid.org/euclid.agt/1513882694

Digital Object Identifier
doi:10.2140/agt.2002.2.297

Mathematical Reviews number (MathSciNet)
MR1917054

Zentralblatt MATH identifier
0991.57004

#### Citation

Rieck, Yo’av; Sedgwick, Eric. Thin position for a connected sum of small knots. Algebr. Geom. Topol. 2 (2002), no. 1, 297--309. doi:10.2140/agt.2002.2.297. https://projecteuclid.org/euclid.agt/1513882694

#### References

• Gerhard Burde and Heiner Zieschang. Knots. Walter de Gruyter & Co., Berlin, 1985.
• M. Culler, C. McA. Gordon, J. Luecke, and P. B. Shalen. Dehn surgery on knots. Ann. of Math., 125:237–300, 1987.
• David Gabai. Foliations and the topology of $3$-manifolds. III. J. Differential Geom., 26(3):479–536, 1987.
• C. McA. Gordon and J. Luecke. Knots are Determined by Their Complements. J. Amer. Math. Soc., 2:371–415, 1989.
• Daniel J. Heath and Tsuyoshi Kobayashi. Essential tangle decomposition from thin position of a link. Pacific J. Math., 179(1):101–117, 1997.
• Kanji Morimoto. Tunnel number and connected sum of knots and links. In Proc. Applied Math Workshop, volume 4, pages 117–128, 1994.
• Kanji Morimoto. On the super additivity of tunnel number of knots. Math. Ann., 317(3):489–508, 2000.
• Yo'av Rieck. Heegaard structures of manifolds in the Dehn filling space. Topology, 39(3):619–641, 2000.
• Yo'av Rieck and Eric Sedgwick. Finiteness results for Heegaard surfaces in surgered manifolds. Comm. Anal. Geom., 9(2):351–367, 2001.
• M. Scharlemann and A. Thompson. Thin position and Heegaard splittings of the $3$-sphere. J. Diff. Geom., 39(2):343–357, 1994.
• Horst Schubert. Über eine numerische Knoteninvariante. Math. Z., 61:245–288, 1954.
• Jennifer Schultens. Additivity of bridge number of knots. Preprint, 2001.
• Abigail Thompson. Thin position and bridge number for knots in the $3$-sphere. Topology, 36(2):505–507, 1997.