1,935 research outputs found
On the Coloring of Pseudoknots
Pseudodiagrams are diagrams of knots where some information about which
strand goes over/under at certain crossings may be missing. Pseudoknots are
equivalence classes of pseudodiagrams, with equivalence defined by a class of
Reidemeister-type moves. In this paper, we introduce two natural extensions of
classical knot colorability to this broader class of knot-like objects. We use
these definitions to define the determinant of a pseudoknot (i.e. the
pseudodeterminant) that agrees with the classical determinant for classical
knots. Moreover, we extend Conway notation to pseudoknots to facilitate the
investigation of families of pseudoknots and links. The general formulae for
pseudodeterminants of pseudoknot families may then be used as a criterion for
p-colorability of pseudoknots.Comment: 22 pages, 24 figure
Purity in categories of sheaves
We consider categorical and geometric purity for sheaves of modules over a
scheme satisfying some mild conditions, both for the category of all sheaves
and for the category of quasicoherent sheaves. We investigate the relations
between these four purities and compute a number of examples, in particular
describing both the geometric and categorical Ziegler spectra for the category
of quasicoherent sheaves over the projective line over a field
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