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
A classical result states that the determinant of an alternating link is equal to the number of spanning trees in a checkerboard graph of an alternating connected projection of the link.We generalize this result to show that the determinant is the alternating sum of the number of quasi-trees of genus j of the dessin of a non-alternating link.Furthermore, we obtain formulas for coefficients of the Jones polynomial by counting quantities on dessins. In particular we will show that the j-th coefficient of the Jones polynomial is given by sub-dessins of genus less or equal to j.
A knot K is called n-adjacent to another knot K if K admits a projection containing n generalized crossings such that changing any 0 < m ≤ n of them yields a projection of K . We apply techniques from the theory of sutured 3-manifolds, Dehn surgery and the theory of geometric structures of 3-manifolds to study the extent to which nonisotopic knots can be adjacent to each other. A consequence of our main result is that if K is n-adjacent to K for all n ∈ ,ގ then K and K are isotopic. This provides a partial verification of the conjecture of V. Vassiliev that finite type knot invariants distinguish all knots. We also show that if no twist about a crossing circle L of a knot K changes the isotopy class of K , then L bounds a disc in the complement of K . This leads to a characterization of nugatory crossings on knots.
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