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A diagrammatic Alexander invariant of tangles STEPHEN BIGELOW
 

Summary: A diagrammatic Alexander invariant of tangles
STEPHEN BIGELOW
Abstract We give a new construction of the one-variable Alexander polyno-
mial of an oriented knot or link, and show that it generalizes to a vector valued
invariant of oriented tangles.
AMS Classification 57M27 ;
Keywords Alexander polynomial, tangle, skein theory, planar algebra.
1 Introduction
The Alexander polynomial is the unique invariant of oriented knots and tangles
that is one for the unknot and satisfies the Alexander-Conway skein relation.
- = (q - q-1
) .
Many other equivalent definitions are known. The aim of this paper is to give yet
another definition of the Alexander polynomial, which we will prove is equivalent
to the above skein theoretic definition.
An advantage of our definition is that it generalizes immediately to give an invari-
ant of oriented tangles. Other generalizations of the Alexander polynomial to tan-
gles have been given in [CT07] and [Arc08]. Their definitions are for the multivari-
able Alexander polynomial, whereas this paper only concerns the single variable
version.

  

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

 

Collections: Mathematics