Geodesic
The diameter of the set of boundary slopes of a knot
Klaff, Ben1 ; Shalen, Peter B2
1Department of Mathematics, University of Texas at Austin, 1 University Station, Austin, TX 78741, USA
2Department of Mathematics, Statistics, and Computer Science (M/C 249), University of Illinois at Chicago, 851 S. Morgan St., Chicago, IL 60607-7045, USA
Algebraic and Geometric Topology, Tome 6 (2006) no. 3, pp. 1095-1112
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Let K be a tame knot with irreducible exterior M(‘K) in a closed, connected, orientable 3–manifold Σ such that π1(Σ) is cyclic. If ∞ is not a strict boundary slope, then the diameter of the set of strict boundary slopes of K, denoted dK, is a numerical invariant of K. We show that either (i) dK ≥ 2 or (ii) K is a generalized iterated torus knot. The proof combines results from Culler and Shalen [Comment. Math. Helv. 74 (1999) 530-547] with a result about the effect of cabling on boundary slopes.

DOI : 10.2140/agt.2006.6.1095
Keywords: knot exterior, strict essential surface, strict boundary slope, diameter, $3$–manifold, cyclic fundamental group, cable knot, generalized iterated torus knot

Klaff, Ben  1   ; Shalen, Peter B  2

1 Department of Mathematics, University of Texas at Austin, 1 University Station, Austin, TX 78741, USA
2 Department of Mathematics, Statistics, and Computer Science (M/C 249), University of Illinois at Chicago, 851 S. Morgan St., Chicago, IL 60607-7045, USA 
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Klaff, Ben; Shalen, Peter B. The diameter of the set of boundary slopes of a knot. Algebraic and Geometric Topology, Tome 6 (2006) no. 3, pp. 1095-1112. doi: 10.2140/agt.2006.6.1095

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