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Abstract
We study representations of the knot groups of twist knots into
SL 2 ( ℂ ) . The set of nonabelian
SL 2 ( ℂ ) representations of a
twist knot
K is described
as the zero set in
ℂ
×
ℂ
of a polynomial
P K ( x , y )
= Q K ( y )
+ x 2 R K ( y )
∈
ℤ [ x , y ] ,
where
x
is the trace of a meridian. We prove some properties of
P K ( x , y ) . In particular,
we prove that
P K ( 2 , y )
∈
ℤ [ y ] is
irreducible over
ℚ .
As a consequence, we obtain an alternative proof of a result of Hoste and Shanahan
that the degree of the trace field is precisely two less than the minimal crossing
number of a twist knot.
Keywords
Chebychev polynomial, nonabelian representation, parabolic
representation, trace field, twist knot
Mathematical Subject Classification 2010
Primary: 57N10
Secondary: 57M25
Milestones
Received: 6 July 2015
Revised: 30 October 2015
Accepted: 3 November 2015
Published: 25 August 2016
Communicated by Jim Hoste