2006/06/19 by Thomas Fiedler, Fiedler, Thomas · 2 citations
Mathematics · #Advanced Combinatorial Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #math.AT #math.GT #msc:57M25
paper · pdf · doi:10.48550/arxiv.math/0606443
48 pages, 42 figures
arxiv created 2006/06/19 · arxiv updated 2009/12/01
Let ϕ: S1× D2→ S1 be the natural projection. An oriented knot K\hookrightarrow V = S1× D2 is called an almost closed braid if the restriction of ϕ to K has exactly two (non-degenerate) critical points (and K is a closed braid if the restriction of ϕ has no critical points at all). We introduce new isotopy invariants for closed braids and almost closed braids in the solid torus V. These invariants refine finite type invariants. They are still calculable with polynomial complexity with respect to the number of crossings of K. Let the solid torus V be standardly embedded in the 3-sphere and let A be the axis of the complementary solid torus S3∖ V. We give examples which show that our invariants can detect non-invertibility of 2-component links K∪ A\hookrightarrow S3. Notice that all quantum link invariants fail to do so and that it is not known wether there are finite type invariants which can detect non-invertibility of 2-component links.