PrefaceAround 1980, W. Thurston proved that every knot complement satisfies the geometrization conjecture: it decomposes into pieces that admit locally homogeneous geometric structures. In addition, he proved that the complement of any non-torus, non-satellite knot admits a complete hyperbolic metric which, by the Mostow-Prasad rigidity theorem, is is necessarily unique up to isometry. As a result, geometric information about a knot complement, such as its volume, gives topological invariants of the knot.Since the mid-1980's, knot theory has also been invigorated by ideas from quantum physics, which have led to powerful and subtle knot invariants, including the Jones polynomial and its relatives, the colored Jones polynomials. Topological quantum field theory predicts that these quantum invariants are very closely connected to geometric structures on knot complements, and particularly to hyperbolic geometry. The volume conjecture of R. Kashaev, H. Murakami, and J. Murakami, which asserts that the volume of a hyperbolic knot is determined by certain asymptotics of colored Jones polynomials, fits into the context of these predictions. Despite compelling experimental evidence, these conjectures are currently verified for only a few examples of hyperbolic knots.This monograph initiates a systematic study of relations between quantum and geometric knot invariants. Under mild diagrammatic hypotheses that arise naturally in the study of knot polynomial invariants (A-or B-adequacy), we derive direct and concrete relations between colored Jones polynomials and the topology of incompressible spanning surfaces in knot and link complements. We prove that the growth of the degree of the colored Jones polynomials is a boundary slope of an essential surface in the knot complement, and that certain coefficients of the polynomial measure how far this surface is from being a fiber in the knot complement. In particular, the surface is a fiber if and only if a certain coefficient vanishes.Our results also yield concrete relations between hyperbolic geometry and colored Jones polynomials: for certain families of links, coefficients of the polynomials determine the hyperbolic volume to within a factor of 4. Our methods here provide a deeper and more intrinsic explanation for similar connections that have been previously observed.Our approach is to generalize the checkerboard decompositions of alternating knots and links. For A-or B-adequate diagrams, we show that the checkerboard knot surfaces are incompressible, and obtain an ideal polyhedral decomposition of their complement. We use normal surface theory to establish a dictionary between the pieces of the JSJ decomposition of the surface complement and the combinatorial structure of certain spines of the checkerboard surface (state graphs). In particular, we give a combinatorial formula 4 for the complexity of the hyperbolic part of the JSJ decomposition (the guts) of the surface complement in terms of the diagram of the knot, and use this to give lower bounds on volumes of several ...

Abstract. Given a hyperbolic 3-manifold with torus boundary, we bound the change in volume under a Dehn filling where all slopes have length at least 2π. This result is applied to give explicit diagrammatic bounds on the volumes of many knots and links, as well as their Dehn fillings and branched covers. Finally, we use this result to bound the volumes of knots in terms of the coefficients of their Jones polynomials.

The Jones polynomial of an alternating link is a certain specialization of the Tutte polynomial of the (planar) checkerboard graph associated to an alternating projection of the link. The Bollobas-Riordan-Tutte polynomial generalizes the Tutte polynomial of planar graphs to graphs that are embedded in closed oriented surfaces of higher genus. In this paper we show that the Jones polynomial of any link can be obtained from the Bollobas-Riordan-Tutte polynomial of a certain oriented ribbon graph associated to a link projection. We give some applications of this approach.Comment: 19 pages, 9 figures, minor change

We construct a polynomial invariant, for links in a large class of rational homology 3-spheres, which generalizes the 2-variable Jones polynomial (HOMFLY). As a consequence, we show that the dual of the HOMFLY skein module of a homotopy 3-sphere is isomorpic to that of the genuine 3-sphere .In order to state our main results we need to introduce some notation: Suppose that M is an orientable 3-manifold. Let π = π 1 (M ) and letπ denote the set of nontrivial conjugancy classes of π. Notice thatπ can be identified with the set of non-trivial free homotopy classes of oriented loops in M . An n-component link is a collection of n unordered oriented circles, tamely and disjointly embedded in M . Hence, a link is homotopically equivalent to an unordered n-tuple of elements inπ ∪ {1}. In every homotopy class of links, we will fix, once and for all, a link CL and call it a trivial link.If CL has k components which are homotopically trivial, our choice will be such that CL = CL * U k , where U k is the standard unlink with k components in a small ball neighborhood disjoint from CL * and U 1 will be abbreviated to U later on. We will denote by CL * the set of all trivial links with none of their components homotopically trivial. It is obvious that CL * is in 1-1 correspondence with unordered n-tuples of elements inπ, for all n > 0.Every link L is homotopic to a certain CL * U k for some CL * ∈ CL * , possibly empty. But the aim of link theory in the 3-manifold M is to understand how two links can differ up to a (tame) isotopy if they are homotopic. Let L be the set of isotopy classes of links in M and let R = C[v ±1 , z ±1 ] be the ring of Laurent polynomials in v and z. A map L → R will be called a link polynomial. Now let z = t 1 2 − t − 1 2 and let I be the ideal of R[t] generated by v − v −1 and t. LetR be the pro-I completion of R[t], i.e. the inverse limit of · · · → R[t]/I n → R[t]/I n−1 → · · · . Theorem A. Let M be a rational homology 3-sphere which is either atoroidal or a Seifert fibered space. Then, there is a unique map J M : L →R satisfying the HOMFLY skein relation v −1 J M (L + ) − vJ M (L − ) = zJ M (L o ) and with given values J M (U ) and J M (CL * ) for every CL * ∈ CL * . Moreover, we may choose J M (U ) and J M (CL * ) in R appropriately such that J M is a link polynomial, i.e. J M (L) ∈ R for every L ∈ L.Here, as usual, the three links L + , L − and L o appeared in the HOMFLY skein relation differ only in a small ball neighborhood in M where, under a suitable projection, they intersect at a positive crossing, a negative crossing, and a smoothing of a crossing, respectively. (1989), 359-389. 121

We use the Jaco-Shalen and Johannson theory of the characteristic submanifold and the Torus theorem (Gabai, Casson-Jungreis) to develop an intrinsic finite tvne theory for knots in irreducible 3-manifolds. We also establish _ __ a relation between finite type knot invariants in 3-manifolds and these in R3. existence of non-trivial finite type invariants for knots in irreducible 3-manifolds. rights reserved 0. INTRODUCTION As an application we obtain the