The Most Refined Invariant of Degree One of Knots and Links in $R^1$-Fibrations Over a Surface

Details The Most Refined Invariant of Degree One of Knots and Links in $R^1$-Fibrations Over a Surface The Most Refined Invariant of Degree One of Knots and Links in $R^1$-Fibrations Over a Surface

1999-06-21

arxiv.org/abs/math/9906137v1

J. Knot Theory Ramifications 7 (1998), no. 2, pp. 257-266

Mathematics

Geometric Topology

9 pages, 7 figures

Scientific paper

As it is well-known, all Vassiliev invariants of degree one of a knot $K\subset R^3$ are trivial. There are nontrivial Vassiliev invariants of degree one, when the ambient space is not $R^3$. Recently, T. Fiedler introduced such invariants of a knot in an $R^1$-fibration over a surface $F$. They take values in the free $Z$-module generated by all the free homotopy classes of loops in $F$. Here, we generalize them to the most refined Vassiliev invariant of degree one. The ranges of values of all these invariants are explicitly described. We also construct a similar invariant of a two-component link in an $\R^1$-fibration. It generalizes the linking number.

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