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Get Continuous Lattices: Proceedings PDF

By B. Banaschewski, R.-E. Hoffmann

ISBN-10: 3540108483

ISBN-13: 9783540108481

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1). 3. The Conway polynomial In what follows we shall usually consider invariants with values in a commutative ring. Of special importance in knot theory are polynomial knot invariants taking values in the rings of polynomials (or Laurent polynomials1) in one or several variables, usually with integer coefficients. Historically, the first polynomial invariant for knots was the Alexander polynomial A(K) introduced in 1928 [Al2]. See [CrF, Lik, Rol] for a discussion of the beautiful topological theory related to the Alexander polynomial.

Kauffman [Ka6], how this definition could be invented. As with the Conway polynomial, the construction given below requires that we consider invariants on the totality of all links, not only knots, because the transformations used may turn a knot diagram into a link diagram with several components. Suppose that we are looking for an invariant of unoriented links, denoted by angular brackets, that has a prescribed behaviour with respect to the resolution of diagram crossings and the addition of a disjoint copy of the unknot: =a L⊔ +b , =c L , where a, b and c are certain fixed coefficients.

These depend on all the data involved: the quantum group, the representation and the R-matrix. 6. Quantum invariants q 1/2 Cy −→ max = V V* ←− max = V* V y • 0y 0y • • q −1/2 y • V ⊗V∗ e1 ⊗ e1 e1 ⊗ e2 e2 ⊗ e1 e2 ⊗ e2 Cy 1y 0y 0y 1y V∗⊗V e1 ⊗ e1 • • e1 ⊗ e2 • e2 ⊗ e1 • e2 ⊗ e2 where {e1 , e2 } is the basis of V ∗ dual to the basis {e1 , e2 } of the space V . We leave to the reader the exercise to check that these operators are consistent with the oriented Turaev moves from page 32. See Exercise 38 for their generalization to slN .

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Continuous Lattices: Proceedings by B. Banaschewski, R.-E. Hoffmann


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