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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 2 RESEARCH BLOG 5/11/04 CORNELL TOPOLOGY FESTIVAL 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.