I Still don't Understand the Alexander Polynomial



2021 04 01

14:30 (GMT+0)


Dror Bar-Natan

Mathematical colloquium in BMSTU

As an algebraic knot theorist, I still don't understand the Alexander
polynomial. There are two conventions as for how to present tangle
theory in algebra: one may name the strands of a tangle, or one may name
their ends. The distinction might seem too minor to matter, yet it leads
to a completely different view of the set of tangles as an algebraic
structure. There are lovely formulas for the Alexander polynomial as
viewed from either perspective, and they even agree where they meet. But
the "strands" formulas know about strand doubling while the "ends" ones
don't, and the "ends" formulas know about skein relations while the
"strands" ones don't. There ought to be a common generalization, but I
don't know what it is.

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