A quantum invariant of knots
| Speaker: | David Jackson |
|---|---|
| Affiliation: | University of Waterloo |
| Room: | Mathematics & Computer Building (MC) 5158 |
Abstract:
A (mathematical) knot is the everyday, perhaps mundane, object but with its two free ends having been smoothly sealed together. Two knots are to be regarded as 'equivalent' if one may be transformed into the other by the smooth operations of stretching or pulling. Thus, mathematically, a knot is an embedding of the unit circle into $\mathbb{R}^3$ considered up to ambient isotopy.
A
classical
question
is
how
to
construct
a
function
$\theta$
that distinguishes
between
knots.
That
is,
for
the
set
$\mathcal{K}$
of
all
knots,
how
to
find
a
suitable
set
$\mathcal{S}$
and
a
function
$\theta\colon\mathcal{K}
\rightarrow\mathcal{S}$
such
that
$$\theta(\mathfrak{s})
\neq
\theta(\mathfrak{t})\quad
\Rightarrow
\quad
\mathfrak{s}
\neq
\mathfrak{t},$$
where
$\mathfrak{s}$
and
$\mathfrak{t}$
are
knots.
Such
a
function
is
called
a
\emph{knot
invariant}.
Not
surprisingly,
as
a
space
embedding
in
another,
knots
and
their
higher
dimensional
analogues
occur
in
an
essential
way
in
other
branches
of
mathematics
and
in
mathematical
physics,
as
the
appearance
of
the
word
"quantum"
in
the
title
suggests.
I
shall
use
Turaev's
Theorem
as
the
theme
for
this
general
talk.
The
idea
is
to
project
a
knot
into
$\mathbb{R}^2$,
equip
the
resulting
image
with
over-
and
under-crossings
to
form
a
\emph{knot
diagram}
(so
that
the
original
knot
may
be
reconstructed),
confine
all
of
the
over-
and
under-crossings
in
the
knot
diagram
to
a
\emph{braid
diagram},
and
then
\emph{close}
the
braid
diagram
by
joining
the
top
and
the
bottom
of
the
braid
diagram
by
parallel
strands.
There
are
two
immediate
questions
about
this
construction:
"How
may
it
be
made
bijective?"
and
"How
does
one
use
it
to
produce
an
actual
invariant?"
In
this
talk,
I
shall
sketch
the
diagrammatic
and
algebraic
ideas
that
lead
to
Turaev's
Theorem,
and
thence
to
the
existence
of
the
Jones
Polynomial.
In
a
later
talk,
I
shall
discuss
Vassiliev
invariants
and
their
connexions
with
combinatorial
aspects
of
Hopf
algebras
and
Lie
algebras,
and
the
association
with
homological
ideas.
On
a
purely
personal
note,
my
interest
in
knots
was
piqued
when
working
on
the
toy
model
of
string
theory
in
the
1990s,
using
graph
embeddings
and
matrix
models,
with
Malcolm
Perry
at
DAMTP
in
Cambridge
and
Terry
Visentin
at
University
of
Winnipeg.
Later,
I
was
also
enticed
by
ramified
covers
of
the
sphere
and
intersection
questions
in
the
moduli
space
of
curves
which,
in
spite
of
the
perhaps
unfamiliar
terminology,
contain
a
lot
of
combinatorics.
The
Goulden-Jackson-Vakil
Conjecture
on
the
existence
of
a
particular
moduli
space
is
a
topic
of
current
interest
in
algebraic
geometry.