arXiv subjects

Jason Callahan

Publications and source records attributed to Jason Callahan.

3 recordsLinked to original sources

arXiv2013

Denominators and Differences of Boundary Slopes for (1,1)-Knots

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.

math.GT↗

arXiv2009

J{\o}rgensen Number and Arithmeticity

A J{\o}rgensen group is a non-elementary Kleinian group that can be generated by two elements for which equality holds in J{\o}rgensen's Inequality. This paper shows that the only torsion-free J{\o}rgensen group is the figure-eight knot group, identifies all non-cocompact arithmetic J{\o}rgensen groups, and establishes a characterization of cocompact arithmetic J{\o}rgensen groups. The paper also defines and computes the J{\o}rgensen number of several non-cocompact Kleinian groups including some two-bridge knot and link groups.

math.GT↗

arXiv2008

Conjugate Generators of Knot and Link Groups

This note shows that if two elements of equal trace (e.g., conjugate elements) generate an arithmetic two-bridge knot or link group, then the elements are parabolic. This includes the figure-eight knot and Whitehead link groups. Similarly, if two conjugate elements generate the trefoil knot group, then the elements are peripheral.

math.GT↗