On spaces of connected graphs II: Relations in the algebra Lambda

Details On spaces of connected graphs II: Relations in the algebra Lambda On spaces of connected graphs II: Relations in the algebra Lambda

2003-01-03

arxiv.org/abs/math/0301019v1

Jour. of Knot Theory and its Ramif. vol. 10, no. 5 (2001), 667-674

Mathematics

Quantum Algebra

8 pages

Scientific paper

The graded algebra Lambda defined by Pierre Vogel is of general interest in the theory of finite-type invariants of knots and of 3-manifolds because it acts on the corresponding spaces of connected graphs subject to relations called IHX and AS. We examine a subalgebra Lambda_0 that is generated by certain elements called t and x_n with n >= 3. Two families of relations in Lambda_0 are derived and it is shown that the dimension of Lambda_0 grows at most quadratically with respect to degree. Under the assumption that t is not a zero divisor in Lambda_0, a basis of Lambda_0 and an isomorphism from Lambda_0 to a sub-ring of Z[t,u,v] is given.

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