1,721,002 research outputs found
The C-polynomial of a knot
No full textIn an earlier paper the first author defined a non-commutative A–polynomial for knots in 3–space, using the colored Jones function. The idea is that the colored Jones function of a knot satisfies a non-trivial linear q–difference equation. Said differently, the colored Jones function of a knot is annihilated by a non-zero ideal of the Weyl algebra which is generalted (after localization) by the non-commutative A–polynomial of a knot. In that paper, it was conjectured that this polynomial (which has to do with repre-sentations of the quantum group Uq.sl2/) specializes at q D 1 to the better known A–polynomial of a knot, which has to do with genuine SL2.C / representations of the knot complement. Computing the non-commutative A–polynomial of a knot is a difficult task which so far has been achieved for the two simplest knots. In the present paper, we introduce the C –polynomial of a knot, along with its non-commutative version, and give an explicit computation for all twist knots. In a forthcoming paper, we will use this information to compute the non-commutative A–polynomial of twist knots. Finally, we formulate a number of conjectures relating the A, the C –polynomial and the Alexander polynomial, all confirmed for the class of twist knots
On Finite Type 3-Manifold Invariants IV: Comparison of Definitions
No full text. The present paper is a continuation of [Ga], [GL1] and [GO]. Using a key Lemma we compare the two currently existing definitions of finite type invariants of oriented integral homology spheres and show that type 3m invariants in the sense of Ohtsuki [Oh] are included in type m invariants in the sense of the first author [Ga]. This partially answers question 1 of [Ga]. We show that type 3m invariants of integral homology spheres in the sense of Ohtsuki map to type 2m invariants of knots in S 3 , thus answering question 2 from [Ga]. Contents 1. Introduction 2 1.1. Definitions 1.2. Statement of the results 1.3. Questions 1.4. Plan of the proof 1.5. Acknowledgement 2. A key Lemma 4 3. Proof of Theorems 1 and 2 6 3.1. Proof of Theorem 1 3.2. Proof of Theorem 2 4. Proof of Theorem 3 12 The authors were partially supported by NSF grants DMS-95-05106 and DMS-93-03489. This and related preprints can also be obtained by accessing the WEB in the address http://www.math. brown.edu/¸stavr..
Whitehead Doubling Persists
No full text. Even though the (untwisted) Whitehead doubling operation kills all known abelian invariants of knots (and makes them topologically slice), we show that it does not kill the rational function that equals to the 2-loop part of the Kontsevich integral. 1. Introduction 1.1. History. It is well-known that to a knot K in an integral homology 3-sphere M one can associate its (symmetrized, normalized) Alexander polynomial #(M,K), (or #(K), in case M is clear) which is an element of the ring #O := Z[t 1 ] Sym 2 where Sym 2 acts by sending t to t -1 . The Alexander polynomial measures classical abelian algebraic topology invariants of knots, namely the order of the torsion module H 1 ( f M , Z) as a Z[t 1 ] module, where f M is the universal abelian cover of M rK. There are several ways to view the Alexander polynomial; the one that we wish to emphasize is the surgical view, see [L, Ro]. Form this point of view, the symmetry and normalization of the Alexander polynomial follow..
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