Higher arity self-distributive operations in Cascades and their cohomology

We investigate constructions of higher arity self-distributive operations, and give relations between cohomology groups corresponding to operations of different arities. For this purpose we introduce the notion of mutually distributive [Formula: see text]-ary operations generalizing those for the binary case, and define a cohomology theory labeled by these operations. A geometric interpretation in terms of framed links is described, with the scope of providing algebraic background of constructing [Formula: see text]-cocycles for framed link invariants. This theory is also studied in the context of symmetric monoidal categories. Examples from Lie algebras, coalgebras and Hopf algebras are given.

The Casson–Walker invariant of 3-manifolds with genus one open book decompositions

In this paper, we give two formulae of values of the Casson–Walker invariant of 3-manifolds with genus one open book decompositions; one is a formula written in terms of a framed link of a surgery presentation of such a 3-manifold, and the other is a formula written in terms of a representation of the mapping class group of a 1-holed torus. For the former case, we compute the invariant through the combinatorial calculation of the degree 1 part of the LMO invariant. For the latter case, we construct a representation of a central extension of the mapping class group through the action of the degree 1 part of the LMO invariant on the space of Jacobi diagrams on two intervals, and compute the invariant as the trace of the representation of a monodromy of an open book decomposition.

Closed, oriented, connected 3-manifolds are subtle equivalence classes of plane graphs

A blink is a plane graph with an arbitrary bipartition of its edges. As a consequence of a recent result of Martelli, it is shown that the homeomorphisms classes of closed oriented 3-manifolds are in 1-1 correspondence with specific classes of blinks. In these classes, two blinks are equivalent if they are linked by a finite sequence of local moves, where each one appears in a concrete list of 64 moves: they are organized in 8 types, each being essentially the same move on 8 simply related configurations. The size of the list can be substantially decreased at the cost of loosing symmetry, just by keeping a very simple move type, the ribbon moves denoted [Formula: see text] (which are in principle redundant). The inclusion of [Formula: see text] implies that all the moves corresponding to plane duality (the starred moves), except for [Formula: see text] and [Formula: see text], are redundant and the coin calculus is reduced to 36 moves on 36 coins. A residual fraction link or a flink is a new object which generalizes blackboard-framed link. It plays an important role in this work. It is in the aegis of this work to find new important connections between 3-manifolds and plane graphs.

Mod2 local invariants of maps between 3-manifolds

This paper refers to the work [V. Goryunov, Local invariants of maps between 3-manifolds, J. Topology 6 (2013) 757–776] on local invariants of maps between 3-manifolds. It is assumed that the manifolds have no boundary, and that the source is compact. In the case when the source and the target are oriented, Goryunov proved that every local order one invariant with integer values can be written as a linear combination of seven basic invariants, and gave a geometrical interpretation for them. When the target is the oriented [Formula: see text], there are further four basic mod2 invariants. One of the mod2 invariants has been provided with a topological interpretation, in terms of the number of components and of the self-linking of a framed link constructed from the cuspidal edge. Here, we show that two further independent linear combinations of the mod2 invariants have a topological interpretation, involving the self-linking number of two curves defined by all irregular points of the critical value set of a generic map from an oriented closed 3-manifold to [Formula: see text].

PL 4-manifolds admitting simple crystallizations: framed links and regular genus

Simple crystallizations are edge-colored graphs representing piecewise linear (PL) 4-manifolds with the property that the 1-skeleton of the associated triangulation equals the 1-skeleton of a 4-simplex. In this paper, we prove that any (simply-connected) PL 4-manifold [Formula: see text] admitting a simple crystallization admits a special handlebody decomposition, too; equivalently, [Formula: see text] may be represented by a framed link yielding [Formula: see text], with exactly [Formula: see text] components ([Formula: see text] being the second Betti number of [Formula: see text]). As a consequence, the regular genus of [Formula: see text] is proved to be the double of [Formula: see text]. Moreover, the characterization of any such PL 4-manifold by [Formula: see text], where [Formula: see text] is the gem-complexity of [Formula: see text] (i.e. the non-negative number [Formula: see text], [Formula: see text] being the minimum order of a crystallization of [Formula: see text]), implies that both PL invariants gem-complexity and regular genus turn out to be additive within the class of all PL 4-manifolds admitting simple crystallizations (in particular, within the class of all “standard” simply-connected PL 4-manifolds).

ON A BASIS FOR THE FRAMED LINK VECTOR SPACE SPANNED BY CHORD DIAGRAMS

In view of the result of Kontsevich, now often called "the fundamental theorem of Vassiliev theory", identifying the graded dual of the associated graded vector space to the space of Vassiliev invariants filtered by degree with the linear span of chord diagrams modulo the "4T-relation" (and in the unframed case, originally considered by Vassiliev, the "1T-" or "isolated chord relation"), it is a problem of some interest to provide a basis for the space of chord diagrams modulo the 4T-relation. We construct the basis for the vector space spanned by chord diagrams with n chords and m link components, modulo 4T relations for n ≤ 5.

INVARIANTS OF KNOTS DERIVED FROM EQUIVARIANT LINKING MATRICES OF THEIR SURGERY PRESENTATIONS

The quantum U(1) invariant of a closed 3-manifold M is defined from the linking matrix of a framed link of a surgery presentation of M. As an equivariant version of it, we formulate an invariant of a knot K from the equivariant linking matrix of a lift of a framed link of a surgery presentation of K. We show that this invariant is determined by the Blanchfield pairing of K, or equivalently, determined by the S-equivalent class of a Seifert matrix of K, and that the "product" of this invariant and its complex conjugation is presented by the Alexander module of K. We present some values of this invariant of some classes of knots concretely.

COMBINATORIAL DEHN–LICKORISH TWISTS AND FRAMED LINK PRESENTATIONS OF 3-MANIFOLDS REVISITED

From a pseudo-triangulation with n tetrahedra T of an arbitrary closed orientable connected 3-manifold (for short, a 3D-space) M3, we present a gem J′, inducing 𝕊3, with the following characteristics: (a) its number of vertices is O(n); (b) it has a set of p pairwise disjoint couples of vertices {ui, vi}, each named a twistor; (c) in the dual (J′)⋆ of J′ a twistor becomes a pair of tetrahedra with an opposite pair of edges in common, and it is named a hinge; (d) in any embedding of (J′)⋆ ⊂ 𝕊3, the ∊-neighborhood of each hinge is a solid torus; (e) these p solid tori are pairwise disjoint; (f) each twistor contains the precise description on how to perform a specific surgery based in a Denh–Lickorish twist on the solid torus corresponding to it; (g) performing all these p surgeries (at the level of the dual gems) we produce a gem G′ with |G′| = M3; (h) in G′ each such surgery is accomplished by the interchange of a pair of neighbors in each pair of vertices: in particular, |V(G′) = |V(J′)|. This is a new proof, based on a linear polynomial algorithm, of the classical theorem of Wallace (1960) and Lickorish (1962) that every 3D-space has a framed link presentation in 𝕊3 and opens the way for an algorithmic method to actually obtaining the link by an O(n2)-algorithm. Actually this has been done and awaits a proper implentation.

THE CASSON–WALKER–LESCOP INVARIANT AND LINK INVARIANTS

Formulas previously presented for the Casson–Walker invariant are generalized to Lescop's extension. These formulas in terms of linking numbers and surgery coefficients compute the change in Lescop's invariant under crossing changes in a framed link presenting a 3-manifold. This leads us to revisit an old formula for a coefficient of the Conway polynomial. Finally we compute Lescop's invariant of several lens spaces.

ON THE LE-MURAKAMI-OHTSUKI INVARIANT IN DEGREE 2 FOR SEVERAL CLASSES OF 3-MANIFOLDS

The 3-manifolds invariant of Le, Murakami and Ohtsuki is the universal finite type invariant for integral homology spheres. It takes values in the graded algebra of trivalent graphs and it is known that its degree one part is essentially the Casson-Walker-Lescop invariant. Here we compute the degree two term for several classes of 3-manifolds. In particular, we give an expression of ω (ML) up to order 2 when MLis the 3-manifold obtained by Dehn surgery along a framed link L with one or two components.