126 research outputs found
Continuous cohomology of topological quandles
A continuous cohomology theory for topological quandles is introduced, and
compared to the algebraic theories. Extensions of topological quandles are
studied with respect to continuous 2-cocycles, and used to show the differences
in second cohomology groups for specific topological quandles. A method of
computing the cohomology groups of the inverse limit is applied to quandles.Comment: 17 page
Algebraic Properties of Quandle Extensions and Values of Cocycle Knot Invariants
Quandle 2-cocycles define invariants of classical and virtual knots, and
extensions of quandles. We show that the quandle 2-cocycle invariant with
respect to a non-trivial -cocycle is constant, or takes some other
restricted form, for classical knots when the corresponding extensions satisfy
certain algebraic conditions. In particular, if an abelian extension is a
conjugation quandle, then the corresponding cocycle invariant is constant.
Specific examples are presented from the list of connected quandles of order
less than 48. Relations among various quandle epimorphisms involved are also
examined
Algebraic Structures Derived from Foams
Foams are surfaces with branch lines at which three sheets merge. They have
been used in the categorification of sl(3) quantum knot invariants and also in
physics. The 2D-TQFT of surfaces, on the other hand, is classified by means of
commutative Frobenius algebras, where saddle points correspond to
multiplication and comultiplication. In this paper, we explore algebraic
operations that branch lines derive under TQFT. In particular, we investigate
Lie bracket and bialgebra structures. Relations to the original Frobenius
algebra structures are discussed both algebraically and diagrammatically.Comment: 11 pages; 14 figure
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