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Register a new userCentrality and the KRH Invariant
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The purpose of this paper is to discuss the categorical structure for a method of defining quantum invariants of knots, links and three-manifolds. These invariants can be defined in terms of right integrals on certain Hopf algebras. We call such an invariant of 3-manifolds a Hennings invariant. The work reported in this paper has its background in previous work of the authors. The present paper gives an abstract description of these structures and shows how the Hopf algebraic image of a knot lies in the center of the corresponding Hopf algebra. The paper also shows how all the axiomatic properties of a quasi-triangular Hopf algebra are involved in the topology via a functor from the Tangle Category to the Diagrammatic Category of a Hopf Algebra. The invariants described in this paper generalize to invariants of rotational virtual knots. The contents of this paper are an update of the original 1998 version published in JKTR.
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We apply Lescops construction of $mathbb{Z}$-equivariant perturbative invariant of knots and 3-manifolds to the explicit equivariant propagator of AL-paths given in arXiv:1403.8030. We obtain an invariant $hat{Z}_n$ of certain equivalence classes of fiberwise Morse functions on a 3-manifold fibered over $S^1$, which can be considered as a higher loop analogue of the Lefschetz zeta function and whose construction will be applied to that of finite type invariants of knots in such a 3-manifold. We also give a combinatorial formula for Lescops equivariant invariant $mathscr{Q}$ for 3-manifolds with $H_1=mathbb{Z}$ fibered over $S^1$. Moreover, surgery formulas of $hat{Z}_n$ and $mathscr{Q}$ for alternating sums of surgeries are given. This gives another proof of Lescops surgery formula of $mathscr{Q}$ for special kind of 3-manifolds and surgeries, which is simple in the sense that the formula is obtained easily by counting certain graphs in a 3-manifold.
We give a generalization of Fukayas Morse homotopy theoretic approach for 2-loop Chern--Simons perturbation theory to 3-valent graphs with arbitrary number of loops at least 2. We construct a sequence of invariants of integral homology 3-spheres with values in a space of 3-valent graphs (Jacobi diagrams or Feynman diagrams) by counting graphs in an integral homology 3-sphere satisfying certain condition described by a set of ordinary differential equations.
In this paper, it is explained that a topological invariant for 3-manifold $M$ with $b_1(M)=1$ can be constructed by applying Fukayas Morse homotopy theoretic approach for Chern--Simons perturbation theory to a local system on $M$ of rational functions associated to the free abelian covering of $M$. Our invariant takes values in Garoufalidis--Rozanskys space of Jacobi diagrams whose edges are colored by rational functions. It is expected that our invariant gives a lot of nontrivial finite type invariants of 3-manifolds.
In this note we present a combinatorial link invariant that underlies some recent stable homotopy refinements of Khovanov homology of links. The invariant takes the form of a functor between two combinatorial 2-categories, modulo a notion of stable equivalence. We also develop some general properties of such functors.
These are the lecture notes for my course at the 2011 Park City Mathematics Graduate Summer School. The first two lectures covered the basics of the Torelli group and the Johnson homomorphism, and the third and fourth lectures discussed the second cohomology group of the level p congruence subgroup of the mapping class group, following my papers The second rational homology group of the moduli space of curves with level structures and The Picard group of the moduli space of curves with level structures.
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