748 research outputs found
A family of pseudo-Anosov braids with large conjugacy invariant sets
We show that there is a family of pseudo-Anosov braids independently
parameterized by the braid index and the (canonical) length whose smallest
conjugacy invariant sets grow exponentially in the braid index and linearly in
the length and conclude that the conjugacy problem remains exponential in the
braid index under the current knowledge.Comment: 16 pages, 6 figure
Grid diagram for singular links
In this paper, we define the set of singular grid diagrams
which provides a unified description for singular links, singular Legendrian
links, singular transverse links, and singular braids. We also classify the
complete set of all equivalence relations on which induce the
bijection onto each singular object. This is an extension of the known result
of Ng-Thurston for non-singular links and braids.Comment: 33 pages, 34 figure
Legendrian singular links and singular connected sums
We study Legendrian singular links up to contact isotopy. Using a special
property of the singular points, we define the singular connected sum of
Legendrian singular links. This concept is a generalization of the connected
sum and can be interpreted as a tangle replacement, which provides a way to
classify Legendrian singular links. Moreover, we investigate several phenomena
only occur in the Legendrian setup
Subdivisional spaces and graph braid groups
We study the problem of computing the homology of the configuration spaces of
a finite cell complex . We proceed by viewing , together with its
subdivisions, as a subdivisional space--a kind of diagram object in a category
of cell complexes. After developing a version of Morse theory for subdivisional
spaces, we decompose and show that the homology of the configuration spaces
of is computed by the derived tensor product of the Morse complexes of the
pieces of the decomposition, an analogue of the monoidal excision property of
factorization homology.
Applying this theory to the configuration spaces of a graph, we recover a
cellular chain model due to \'{S}wi\k{a}tkowski. Our method of deriving this
model enhances it with various convenient functorialities, exact sequences, and
module structures, which we exploit in numerous computations, old and new.Comment: 71 pages, 15 figures. Typo fixed. May differ slightly from version
published in Documenta Mathematic
On folded cluster patterns of affine type
A cluster algebra is a commutative algebra whose structure is decided by a
skew-symmetrizable matrix or a quiver. When a skew-symmetrizable matrix is
invariant under an action of a finite group and this action is admissible, the
folded cluster algebra is obtained from the original one. Any cluster algebra
of non-simply-laced affine type can be obtained by folding a cluster algebra of
simply-laced affine type with a specific -action. In this paper, we study
the combinatorial properties of quivers in the cluster algebra of affine type.
We prove that for any quiver of simply-laced affine type, -invariance and
-admissibility are equivalent. This leads us to prove that the set of
-invariant seeds forms the folded cluster pattern.Comment: 29 page
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