Summary: A diagrammatic Alexander invariant of tangles
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
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 in-
variant of oriented tangles. Other generalizations of the Alexander polynomial to
tangles have been given in [CT07] and [Arc08]. Their definitions are for the multi-
variable Alexander polynomial, whereas this paper only concerns the single vari-
able version. I hope the definition in this paper can be extended to a multivariable