040512 by shimeiyan

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									                      RESEARCH BLOG 5/11/04
                    CORNELL TOPOLOGY FESTIVAL



   Last weekend was the Cornell Topology Festival. Kronheimer talked
about his work with Mrowka solving property P . In fact, they prove
that if K ⊂ S 3 is a knot, then π1 (K(1/n)) has a non-trivial representa-
tion to SU(2), n = 0 (where K(1/n) means 1/n Dehn surgery on K).
This implies that the character variety of π1 (S 3 − K) → SL2(C) is non-
trivial, since SU(2) ⊂ SL(2, C). To see this, note that H1(K(1/n)) = 0.
Thus, any non-trivial representation π1 (K(1/n)) → SL2 (C) must have
non-cyclic image. This implies that the character variety of π1 (S 3 − K)
distinguishes K from the unknot U , since π1 (S 3 − U ) = Z. This was
known for hyperbolic and torus knot complements, but not for general
non-trivial knots (which must be satellites).
   In fact, this result implies that the A-polynomial distinguishes non-
trivial knots from the unknot. Nathan Dunfield and Stavros Garoufa-
lidis are writing up this result, as well as Boyer and Zhang indepen-
dently. The A-polynomial was defined by Cooper, Culler, Gillet and
Shalen [1]. Given a representation of π1 (S 3 − K) → SL(2, C), one may
normalize the meridian and longitude to be matrices of the form
                           µ α     λ β
                              −1 ,       .
                           0 µ     0 λ−1
The (closure of the) set of possible values of (µ, λ) ∈ C∗ × C∗ forms an
algebraic variety V . An algebraic variety in C2 must be either a finite
collection of points, everything, or a surface cut out by a single polyno-
mial. For a hyperbolic manifold, this polynomial is non-trivial, and is
called the A-polynomial. It turns out that the variety V cannot be all
of C2 , as then every slope would be the boundary slope for an incom-
pressible surface, violating a theorem of Hatcher (which says that there
are at most finitely many boundary slopes). But a priori, V could be a
finite collection of points. If one performs p Dehn surgery on K, then
                                              q
                                                           1
the parameters must be of the form µp λq = ±1. For n surgery, this
                                    1
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means that there must be points (µ, λ) ∈ V of the form µ = ±λ−n , for
all n ∈ Z−{0}. The only way that V could be finite and have this possi-
ble is if for (µ, λ) ∈ V corresponding to reps. π1 (K(1/n)) → SU(2), λ is
a root of unity, and µ is a power of λ. So there is an m such that λm = 1,
for any (µ, λ) ∈ V corresponding to a rep. π1 (K(1/n)) → SU(2). But
then for the non-trivial rep π1 (K(1/m)) → SU(2), there must be a
point of the form (1, λ) ∈ V , which gives a non-trivial representation of
π1 (K(1/0)) = 1 → SL(2, C), a contradiction. Thus, the variety must
be infinite, and the A-polynomial is non-trivial.
   John Etnyre mentioned to me that Lenny Ng has a new polynomial
invariant of knots, which comes from his combinatorial invariant which
is conjectured to be equivalent to an analytic invariant coming from
contact homology (Etnyre and his collaborators are near to showing
this equivalence, according to John, see blog 6/25/03 ). Apparently, in
Ng’s invariant, there are two free variables, corresponding to generators
for the homology of the contact torus associated to a knot. Thus, one
could choose these to correspond to the meridian and longitude of the
knot. For each choice of generator, one gets a differential, which is non-
linear. The set of such choices for which the differential is linearizable
turn out to form a variety (I don’t claim to understand this, this is just
what John told me!). From this variety, Ng can recover the Alexander
polynomial. This sounds suspiciously like the A-polynomial, since the
Alexander polynomial can be recovered from the component of the
character variety corresponding to reducible representations (these are
solvable, and the Alexander polynomial describes the solvable quotient
of π1 (S 3 −K)). It would be remarkable if these turn out to be equivalent
invariants.
   The A-polynomial is also conjectured to be determined by the col-
ored Jones polynomials (see blog 6/25/03 for some discussion). Garo-
ufalidis and Le have shown that the colored Jones polynomials J K (n)
are q-holonomic (as described in a talk by Garoufalidis at Trends in
3-manifolds at UQAM in Montreal the week before). To describe what
this means, let f : N → Z[q, q −1]. There are operators (Qf )(n) =
q n f (n), (Ef )(n) = f (n + 1). Then EQ = qQE, so it is natural to de-
fine the Weyl algebra A = Z[q, q −1] Q, E /{EQ = qQE}. Their result
                 RESEARCH BLOG 5/11/04                        CORNELL TOPOLOGY FESTIVAL                           3




is that there is a polynomial P ∈ A such that P (JK ) = 0. By inverting
Q, one gets an algebra which is a principle ideal domain, so P may be
uniquely identified. Specializing at q = 1 and changing coordinates,
Garoufalidis conjectures that one obtains the A-polynomial.
   There also appears to be connections between the Heegaard Floer
homology invariants of Oszvath and Szabo, and the Khovanov homol-
ogy invariants, which has been at least formally realized in work of
Oszvath-Szabo and Rasmussen (see blog 3/10/04 for some discussion
of this). An approach to relating these is given by a new invariant
of Seidel and Smith, which aims to give a symplectic topology defi-
nition of Khovanov homology, and thus for the Jones polynomial. It
seems as if the time is ripe for many different invariants of knots and
3-manifolds to be related, and possibly unified into a powerful invariant
which contains both topological, geometric, and quantum information.
                                              References
[1] D. Cooper, M. Culler, H. Gillet, D. D. Long, and P. B. Shalen. Plane curves associated to character varieties of
    3-manifolds. Invent. Math., 118(1):47–84, 1994.

								
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