arXiv subjects

Jan Kneissler

Publications and source records attributed to Jan Kneissler.

3 recordsLinked to original sources

arXiv2003

On spaces of connected graphs I: Properties of Ladders

We examine spaces of connected tri-/univalent graphs subject to local relations which are motivated by the theory of Vassiliev invariants. It is shown that the behaviour of ladder-like subgraphs is strongly related to the parity of the number of rungs: there are similar relations for ladders of even and odd lengths, respectively. Moreover, we prove that - under certain conditions - an even number of rungs may be transferred from one ladder to another.

math.QA↗

arXiv2003

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

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.

math.QA↗

arXiv2003

On spaces of connected graphs III: The Ladder Filtration

A new filtration of the spaces of tri-/univalent graphs B_m^u that occur in the theory of finite-type invariants of knots and 3-manifolds is introduced. Combining the results of the two preceding articles, the quotients of this filtration are modeled by spaces of graphs with two types of edges and four types of vertices, and an upper bound for dim B_m^u in terms of the dimensions of the filtration quotients is given. The degree m up to which B_m^u is known is raised by two for u=0 and u=2.

math.QA↗