7,652 research outputs found
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
Quantization of linear Poisson structures and degrees of maps
Kontsevich's formula for a deformation quantization of Poisson structures
involves a Feynman series of graphs, with the weights given by some complicated
integrals (using certain pullbacks of the standard angle form on a circe). We
explain the geometric meaning of this series as degrees of maps of some grand
configuration spaces; the associativity proof is also interpreted in purely
homological terms. An interpretation in terms of degrees of maps shows that any
other 1-form on the circle also leads to a *-product and allows one to compare
these products.Comment: An extended and modified version; 18 pages, 10 figures. To appear in
Let. Math. Phy
Alexander-Conway invariants of tangles
We consider an algebra of (classical or virtual) tangles over an ordered
circuit operad and introduce Conway-type invariants of tangles which respect
this algebraic structure. The resulting invariants contain both the
coefficients of the Conway polynomial and the Milnor's mu-invariants of string
links as partial cases. The extension of the Conway polynomial to virtual
tangles satisfies the usual Conway skein relation and its coefficients are GPV
finite type invariants. As a by-product, we also obtain a simple representation
of the braid group which gives the Conway polynomial as a certain twisted
trace.Comment: 14 pages, many figure
Skein relations for Milnor's mu-invariants
The theory of link-homotopy, introduced by Milnor, is an important part of
the knot theory, with Milnor's mu-bar-invariants being the basic set of
link-homotopy invariants. Skein relations for knot and link invariants played a
crucial role in the recent developments of knot theory. However, while skein
relations for Alexander and Jones invariants are known for quite a while, a
similar treatment of Milnor's mu-bar-invariants was missing. We fill this gap
by deducing simple skein relations for link-homotopy mu-invariants of string
links.Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol5/agt-5-58.abs.htm
Cubic complexes and finite type invariants
Cubic complexes appear in the theory of finite type invariants so often that
one can ascribe them to basic notions of the theory. In this paper we begin the
exposition of finite type invariants from the `cubic' point of view. Finite
type invariants of knots and homology 3-spheres fit perfectly into this
conception. In particular, we get a natural explanation why they behave like
polynomials.Comment: Published by Geometry and Topology Monographs at
http://www.maths.warwick.ac.uk/gt/GTMon4/paper14.abs.htm
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