Geodesic
Small genus knots in lens spaces have small bridge number
Baker, Kenneth L1
1School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0160, USA
Algebraic and Geometric Topology, Tome 6 (2006) no. 4, pp. 1519-1621
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In a lens space X of order r a knot K representing an element of the fundamental group π1X≅ℤ∕rℤ of order s ≤ r contains a connected orientable surface S properly embedded in its exterior X − N(K) such that ∂S intersects the meridian of K minimally s times. Assume S has just one boundary component. Let g be the minimal genus of such surfaces for K, and assume s ≥ 4g − 1. Then with respect to the genus one Heegaard splitting of X, K has bridge number at most 1.

DOI : 10.2140/agt.2006.6.1519
Keywords: (1,1)–knots, Berge knots, bridge position, lens space, Scharlemann cycle, thin position

Baker, Kenneth L  1

1 School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0160, USA 
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Baker, Kenneth L. Small genus knots in lens spaces have small bridge number. Algebraic and Geometric Topology, Tome 6 (2006) no. 4, pp. 1519-1621. doi: 10.2140/agt.2006.6.1519

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