6,875 research outputs found
Fusion and fission in graph complexes
We analyze a functor from cyclic operads to chain complexes first considered
by Getzler and Kapranov and also Markl. This functor is a generalization of the
graph homology considered by Kontsevich, which was defined for the three
operads Comm, Assoc, and Lie. More specifically we show that these chain
complexes have a rich algebraic structure in the form of families of operations
defined by fusion and fission. These operations fit together to form
uncountably many Lie-infinity and co-Lie-infinity structures.
In particular, the chain complexes have a bracket and cobracket which are
compatible in the Lie bialgebra sense on a certain natural subcomplex.Comment: This is the final version. The published version, which is slightly
different, is available at http://nyjm.albany.edu:8000/PacJ/2003/v209-2.ht
Homotopy approximations to the space of knots, Feynman diagrams, and a conjecture of Scannell and Sinha
Scannell and Sinha considered a spectral sequence to calculate the rational
homotopy groups of spaces of long knots in n-dimensional Euclidean space, for n
greater than or equal to 4. At the end of their paper they conjecture that when
n is odd, the terms on the antidiagonal on the second page precisely give the
space of primitive Feynman diagrams related to the theory of Vassiliev
invariants. In this paper we prove that conjecture. This has the application
that the path components of the terms of the Taylor tower for the space of
classical long knots are in one-to-one correspondence with quotients of the
module of Feynman diagrams, even though the Taylor tower does not actually
converge. This provides strong evidence that the stages of the Taylor tower
give rise to universal Vassiliev knot invariants in each degree.Comment: Published version. Cleaned up some expositio
Extending partial isometries of generalized metric spaces
We consider generalized metric spaces taking distances in an arbitrary
ordered commutative monoid, and investigate when a class of
finite generalized metric spaces satisfies the Hrushovski extension property:
for any there is some such that is a
subspace of and any partial isometry of extends to a total isometry of
. Our main result is the Hrushovski property for the class of finite
generalized metric spaces over a semi-archimedean monoid . When
is also countable, this can be used to show that the isometry
group of the Urysohn space over has ample generics. Finally, we
prove the Hrushovski property for classes of integer distance metric spaces
omitting triangles of uniformly bounded odd perimeter. As a corollary, given
odd , we obtain ample generics for the automorphism group of the
universal, existentially closed graph omitting cycles of odd length bounded by
.Comment: 12 pages, final version incorporating referee comment
A knot bounding a grope of class n is n/2-trivial
In this article it is proven that if a knot, K, bounds an imbedded grope of
class n, then the knot is n/2-trivial in the sense of Gusarov and Stanford.
That is, all type n/2 invariants vanish on K. We also give a simple way to
construct all knots bounding a grope of a given class. It is further shown that
this result is optimal in the sense that for any n there exist gropes which are
not n/2+1- trivial.Comment: 32 pages, 25 figures, additional reference material adde
Forking and dividing in Henson graphs
For , define to be the theory of the generic -free graph,
where is the complete graph on vertices. We prove a graph theoretic
characterization of dividing in , and use it to show that forking and
dividing are the same for complete types. We then give an example of a forking
and nondividing formula. Altogether, provides a counterexample to a
recent question of Chernikov and Kaplan.Comment: 11 page
A remark on strict independence relations
We prove that if is a complete theory with weak elimination of
imaginaries, then there is an explicit bijection between strict independence
relations for and strict independence relations for . We use
this observation to show that if is the theory of the Fra\"{i}ss\'{e} limit
of finite metric spaces with integer distances, then has more
than one strict independence relation. This answers a question of Adler [1,
Question 1.7].Comment: 9 pages, to appear in Archive for Mathematical Logi
Ornate necklaces and the homology of the genus one mapping class group
According to seminal work of Kontsevich, the unstable homology of the mapping
class group of a surface can be computed via the homology of a certain lie
algebra. In a recent paper, S. Morita analyzed the abelianization of this lie
algebra, thereby constructing a series of candidates for unstable classes in
the homology of the mapping class group. In the current paper, we show that
these cycles are all nontrivial, representing degree 4k+1 homology classes in
the homology of the mapping class group of a genus one surface with 4k+1
punctures
The Lie Lie algebra
We study the abelianization of Kontsevich's Lie algebra associated with the
Lie operad and some related problems. Calculating the abelianization is a
long-standing unsolved problem, which is important in at least two different
contexts: constructing cohomology classes in
and related groups as well as studying the higher order Johnson homomorphism of
surfaces with boundary. The abelianization carries a grading by "rank," with
previous work of Morita and Conant-Kassabov-Vogtmann computing it up to rank
. This paper presents a partial computation of the rank part of the
abelianization, finding lots of irreducible -representations with
multiplicities given by spaces of modular forms. Existing conjectures in the
literature on the twisted homology of imply that
this gives a full account of the rank part of the abelianization in even
degrees.Comment: Version 3 incorporates several suggestions from the referee. The
abstract and introduction have been rewritten to be more user-friendly. To
appear in Quantum Topolog
Chirality and the Conway polynomial
In recent work with J.Mostovoy and T.Stanford,the author found that for every
natural number n, a certain polynomial in the coefficients of the Conway
polynomial is a primitive integer-valued degree n Vassiliev invariant, but that
modulo 2, it becomes degree n-1. The conjecture then naturally suggests itself
that these primitive invariants are congruent to integer-valued degree n-1
invariants. In this note, the consequences of this conjecture are explored.
Under an additional assumption, it is shown that this conjecture implies that
the Conway polynomial of an amphicheiral knot has the property that
C(z)C(iz)C(z^2) is a perfect square inside the ring of power series with
integer coefficients, or, equivalently, the image of C(z)C(iz)C(z^2) is a
perfect square inside the ring of polynomials with Z_4 coefficients. In fact,
it is probably the case that the Conway polynomial of an amphicheiral knot
always can be written as f(z)f(-z) for some polynomial f(z) with integer
coefficients, and this actually implies the above "perfect squares" conditions.
Indeed, by work of Kawauchi and Hartley, this is known for all negative
amphicheiral knots and for all strongly positive amphicheiral knots. In general
it remains unsolved, and this paper can be seen as some evidence that it is
indeed true in general. [Added 2/22/12: Of note is the recent paper
arXiv:1106.5634v1 [math.GT] by Ermotti, Hongler and Weber, which finds a
counterexample to the conjecture that all Conway polynomials of amphicheiral
knots are of the form f(z)f(-z). Intriguingly, their main example still
satisfies C(z)C(iz)C(z^2)=f(z)^2 for a power series f(z), making the main
conjecture of the present paper that much more compelling, in the author's
opinion.]Comment: This now closely approximates the published versio
The Johnson Cokernel and the Enomoto-Satoh invariant
We study the cokernel of the Johnson homomorphism for the mapping class group
of a surface with one boundary component. A graphical trace map simultaneously
generalizing trace maps of Enomoto-Satoh and Conant-Kassabov-Vogtmann is given,
and using technology from the author's work with Kassabov and Vogtmann, this is
is shown to detect a large family of representations which vastly generalizes
series due to Morita and Enomoto-Satoh. The Enomoto-Satoh trace is the rank 1
part of the new trace. The rank 2 part is also investigated.Comment: Added a reference to forthcoming work with Kassabo
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