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An almost-integral universal Vassiliev invariant of knots. (English) Zbl 1002.57023

The author observes similarities between the Kontsevich integral of a knot and the Chern character and then defines an analogue of the total Chern class \(c(K)\) for a knot \(K\) which is constructed from the Kontsevich integral. The motivation for doing this is the fact that the total Chern class takes values in the cohomology ring over the integers, while the Chern character is defined over the rationals. So it is natural to expect that \(c(K)\) would be integral. It is proved that the integrality holds at the level of an irreducible representation \(\rho\) of a simple Lie algebra, i.e., \(c(K)\) evaluated on the weight system corresponding to \(\rho\) is integral. It is also proved that the integrality does not hold at the diagram level; if \(K\) is the trefoil knot, then \(c(K)\) cannot be expressed as an integral linear combination of diagrams.

MSC:

57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57R20 Characteristic classes and numbers in differential topology
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
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