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A DIAGRAMMATIC DESCRIPTION OF THE MULTIVARIATE ALEXANDER POLYNOMIAL
 

Summary: A DIAGRAMMATIC DESCRIPTION OF THE MULTIVARIATE
ALEXANDER POLYNOMIAL
K. GRACE KENNEDY
1. Introduction
From their introduction in Jones's Planar Algebras I, planar algebras have been
linked to knot invariants [12]. Here we present a planar algebra which can be used
to describe the multivariate Alexander polynomial used by Bigelow for the single
variable Alexander polynomial. Alexander discovered what would become known as
the Alexander polynomial of a knot and published it in his 1928 paper Topological
Invariants of Knots and Links [2]. In 1969, Conway discovered the skein relation
- = (q - q-1
)
which could be used for both identifying when a knot invariant was equivalent to
Alexander's and for calculating the invariant, by making local changes to crossings.
A year later, Conway discovered the Conway potential function, or the multivariate
Alexander polynomial [7], which is more effective at distinguishing links than the
single variable invariant. In 1993, Murakami published a list of axioms for the
multivariate Alexander polynomial in his paper A State Model for the Multi-variable
Alexander Polynomial [14]. This contribution was analogous to the discovery of
the skein relation for the single variable version. One could then determine if a

  

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

 

Collections: Mathematics