Denominators and Differences of Boundary Slopes for (1,1)-Knots
Geometric Topology
2014-07-25 v2
Abstract
We show that every nonzero integer occurs in the denominator of a boundary slope for infinitely many (1,1)-knots and that infinitely many (1,1)-knots have boundary slopes of arbitrarily small difference. Specifically, we prove that for any integers m, n > 1 with n odd the exterior of the Montesinos knot K(-1/2, m/(2m \pm 1), 1/n) in S^3 contains an essential surface with boundary slope r = 2(n-1)^2/n if m is even and 2(n+1)^2/n if m is odd. If n > 4m, we prove that K(-1/2, m/(2m+1), 1/n) also has a boundary slope whose difference with r is (8m-2)/(n^2-4mn+n), which decreases to 0 as n increases. All of these knots are (1,1)-knots.
Cite
@article{arxiv.1301.6298,
title = {Denominators and Differences of Boundary Slopes for (1,1)-Knots},
author = {Jason Callahan},
journal= {arXiv preprint arXiv:1301.6298},
year = {2014}
}
Comments
11 pages, 5 figures. Results added and generalized, paper restructured and exposition improved