53 research outputs found
An sl_n stable homotopy type for matched diagrams
There exists a simplified Bar-Natan Khovanov complex for open 2-braids. The
Khovanov cohomology of a knot diagram made by gluing tangles of this type is
therefore often amenable to calculation. We lift this idea to the level of the
Lipshitz-Sarkar stable homotopy type and use it to make new computations.
Similarly, there exists a simplified Khovanov-Rozansky sl_n complex for open
2-braids with oppositely oriented strands and an even number of crossings.
Diagrams made by gluing tangles of this type are called matched diagrams, and
knots admitting matched diagrams are called bipartite knots. To a pair
consisting of a matched diagram and a choice of integer n >= 2, we associate a
stable homotopy type. In the case n = 2 this agrees with the Lipshitz-Sarkar
stable homotopy type of the underlying knot. In the case n >= 3 the cohomology
of the stable homotopy type agrees with the sl_n Khovanov-Rozansky cohomology
of the underlying knot.
We make some consistency checks of this sl_n stable homotopy type and show
that it exhibits interesting behaviour. For example we find a CP^2 in the sl_3
type for some diagram, and show that the sl_4 type can be interesting for a
diagram for which the Lipshitz-Sarkar type is a wedge of Moore spaces.Comment: 62 pages, color figure
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