2,250 research outputs found
Homological algebra related to surfaces with boundary
In this article we describe an algebraic framework which can be used in three
related but different contexts: string topology, symplectic field theory, and
Lagrangian Floer theory of higher genus. It turns out that the relevant
algebraic structure for all three contexts is a homotopy version of involutive
bi-Lie algebras, which we call IBL-algebras.Comment: 127 pages, 22 figures. Some references added in version 2. Fixed a
tex problem in version
Diassociative algebras and Milnor's invariants for tangles
We extend Milnor's mu-invariants of link homotopy to ordered (classical or
virtual) tangles. Simple combinatorial formulas for mu-invariants are given in
terms of counting trees in Gauss diagrams. Invariance under Reidemeister moves
corresponds to axioms of Loday's diassociative algebra. The relation of tangles
to diassociative algebras is formulated in terms of a morphism of corresponding
operads.Comment: 17 pages, many figures; v2: several typos correcte
Kitaev lattice model for bicrossproduct Hopf algebras and tensor network representation
Kitaev's lattice models are usually defined as representations of the
Drinfeld quantum double. We propose a new version based on Majid's
bicrossproduct quantum group. Given a Hopf algebra , we show that a
triangulated oriented surface defines a representation of the bicrossproduct
quantum group . Even
though the bicrossproduct has a more complicated and entangled coproduct, the
construction of this new model is relatively natural as it relies on the use of
the covariant Hopf algebra actions. We obtain an exactly solvable Hamiltonian
for the model and provide a definition of the ground state in terms of a tensor
network representation.Comment: 34 page
sl(3) link homology
We define a bigraded homology theory whose Euler characteristic is the
quantum sl(3) link invariant.Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-45.abs.htm
A New Matrix-Tree Theorem
The classical Matrix-Tree Theorem allows one to list the spanning trees of a
graph by monomials in the expansion of the determinant of a certain matrix. We
prove that in the case of three-graphs (that is, hypergraphs whose edges have
exactly three vertices) the spanning trees are generated by the Pfaffian of a
suitably defined matrix. This result can be interpreted topologically as an
expression for the lowest order term of the Alexander-Conway polynomial of an
algebraically split link. We also prove some algebraic properties of our
Pfaffian-tree polynomial.Comment: minor changes, 29 pages, version accepted for publication in Int.
Math. Res. Notice
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