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Remarks on the Apolynomial of a Knot. D. COOPER and D.D. LONG

Summary: Remarks on the A­polynomial of a Knot.
Math Dept., U.C.S.B., CA93106, U.S.A.
This paper reviews the two variable polynomial invariant of knots defined
using representations of the fundamental group of the knot complement into
SL2C: The slopes of the sides of the Newton polygon of this polynomial are
boundary slopes of incompressible surfaces in the knot complement. The
polynomial also contains information about which surgeries are cyclic, and
about the shape of the cusp when the knot is hyperbolic. We prove that
at least some mutants have the same polynomial, and that most untwisted
doubles have non­trivial polynomial. We include several open questions.
Keywords: Knot, polynomial, boundary slope, Newton polygon, surgery.
1 Introduction
In this paper we review the two­variable A­polynomial for a knot which was intro­
duced in [3]. Many interesting features concerned with the geometry and topology
of the knot complement are reflected in this polynomial. For example the boundary
slopes of some, or possibly all, of the incompressible embedded surfaces are coded
by it. In the case that the knot is hyperbolic, information about the cusp shape
is in this polynomial. Under certain conditions one may deduce that a knot has


Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara


Collections: Mathematics