We define an invariant $\nabla_G(M)$ of pairs M,G, where M is a 3-manifold obtained by surgery on some framed link in the cylinder $S\times I$, S is a connected surface with at least one boundary component, and G is a fatgraph spine of S. In effect, $\nabla_G$ is the composition with the $ι_n$ maps of Le-Murakami-Ohtsuki of the link invariant of Andersen-Mattes-Reshetikhin computed relative to choices determined by the fatgraph G; this provides a basic connection between 2d geometry and 3d quantum topology. For each fixed G, this invariant is shown to be universal for homology cylinders, i.e., $\nabla_G$ establishes an isomorphism from an appropriate vector space $\bar{H}$ of homology cylinders to a certain algebra of Jacobi diagrams. Via composition $\nabla_{G'}\circ\nabla_G^{-1}$ for any pair of fatgraph spines G,G' of S, we derive a representation of the Ptolemy groupoid, i.e., the combinatorial model for the fundamental path groupoid of Teichmuller space, as a group of automorphisms of this algebra. The space $\bar{H}$ comes equipped with a geometrically natural product induced by stacking cylinders on top of one another and furthermore supports related operations which arise by gluing a homology handlebody to one end of a cylinder or to another homology handlebody. We compute how $\nabla_G$ interacts with all three operations explicitly in terms of natural products on Jacobi diagrams and certain diagrammatic constants. Our main result gives an explicit extension of the LMO invariant of 3-manifolds to the Ptolemy groupoid in terms of these operations, and this groupoid extension nearly fits the paradigm of a TQFT. We finally re-derive the Morita-Penner cocycle representing the first Johnson homomorphism using a variant/generalization of $\nabla_G$.
We introduce an invariant for trivalent fatgraph spines of a once bordered surface, which takes values in the first homology of the surface. This invariant is the secondary object coming from two 1-cocycles on the dual fatgraph complex, one introduced by Morita and Penner in 2008, and the other by Penner, Turaev, and the author in 2013. We present an explicit formula for this invariant and investigate its properties. We also show that the mod 2 reduction of the invariant is the difference of naturally defined two spin structures on the surface.
The mapping class group of a surface with one boundary component admits numerous interesting representations including as a group of automorphisms of a free group and as a group of symplectic transformations. Insofar as the mapping class group can be identified with the fundamental group of Riemann's moduli space, it is furthermore identified with a subgroup of the fundamental path groupoid upon choosing a basepoint. A combinatorial model for this, the mapping class groupoid, arises from the invariant cell decomposition of Teichmüller space, whose fundamental path groupoid is called the Ptolem…
In this paper, we define finite type invariants for cyclic equivalence classes of nanophrases and construct the universal ones. Also, we identify the universal finite type invariant of degree 1 essentially with the linking matrix. It is known that extended Arnold's basic invariants to signed words are finite type invariants of degree 2, by Fujiwara. We give another proof of this result and show that those invariants do not provide the universal one of degree 2.
In this paper, it is shown that there are no nonconstant Goussarov-Polyak-Viro finite-type invariants that are invariant under the virtualization move. As an immediate corollary, we obtain the theorem which states none of the Birman coefficients of the Jones-Kauffman polynomial are of GPV finite type.
Homotopy classes of nanowords and nanophrases are combinatorial generalizations of virtual knots and links. Goussarov, Polyak and Viro defined finite type invariants for virtual knots and links via semi-virtual crossings. We extend their definition to nanowords and nanophrases. We study finite type invariants of low degrees. In particular, we show that the linking matrix and T invariant defined by Fukunaga are finite type of degree one and degree two respectively. We also give a finite type invariant of degree 4 for open homotopy of Gauss words.
We extend the theory of Vassiliev (or finite type) invariants for knots to knotoids using two different approaches. Firstly, we take closures on knotoids to obtain knots and we use the Vassiliev invariants for knots, proving that these are knotoid isotopy invariant. Secondly, we define finite type invariants directly on knotoids, by extending knotoid invariants to singular knotoid invariants via the Vassiliev skein relation. Then, for spherical knotoids we show that there are non-trivial type-1 invariants, in contrast with classical knot theory where type-1 invariants vanish. We give a complet…
We generalize three invariants, first discovered by A. Henrich, to the long and/or framed virtual knot case. These invariants are all finite-type invariants of order one, and include a universal one. The generalization will require us to extend the notion of a based matrix of a virtual string, first introduced by V. Turaev and later generalized by Henrich, to the long and framed cases.
We provide a way to produce knots in $S^3$ from signed chord diagrams, and prove that every knot can be produced in this way. Using these diagrams, we generalize the fundamental theorem of finite type invariants. We also provide moves for the diagrams so that any two diagrams for the same knot are connected by a sequence of moves.
We define an invariant of tangles and framed tangles given a finite crossed module and a pair of functions, called a Reidemeister pair, satisfying natural properties. We give several examples of Reidemeister pairs derived from racks, quandles, rack and quandle cocycles, and central extensions of groups. We prove that our construction includes all rack and quandle cohomology (framed) link invariants, as well as the Eisermann invariant of knots. We construct a class of Reidemeister pairs which constitute a lifting of the Eisermann invariant, and show through an example that this class is strictl…
Based on a recently introduced by the author notion of {\em parity}, in the present paper we construct a sequence of invariants (indexed by natural numbers $m$) of long virtual knots, valued in certain simply-defined group ${\tilde G}_{m}$ (the Cayley graphs of these groups are represented by grids in the $(m+1)$-space); the conjugacy classes of elements of $G_{m}$ play the role of invariants of {\em compact} virtual knots. By construction, all invariants do not change under {\em virtualization}. Factoring the group algebra of the corresponding group by certain polynomial relations leads to fi…
We construct an infinite commutative lattice of groups whose dual spaces give Kauffman finite-type invariants of long virtual knots. The lattice is based "horizontally" upon the Polyak algebra and extended "vertically" using Manturov's functorial map $f$. For each $n$, the $n$-th vertical line in the lattice contains an infinite dimensional subspace of Kauffman finite-type invariants of degree $n$. Moreover, the lattice contains infinitely many inequivalent extensions of the Conway polynomial to long virtual knots, all of which satisfy the same skein relation. Bounds for the rank of each group…
We investigate how one can twist L^2-invariants such as L^2-Betti numbers and L^2-torsion with finite-dimensional representations. As a special case we assign to the universal covering of a finite connected CW-complex X together with an element phi in H^1(X;R) a phi-twisted L^2-torsion function from R^{>0} to R, provided that the fundamental group of X is residually finite and its universal covering is L^2-acyclic.
We define ambient isotopy invariants of oriented knots and links using the counting invariants of framed links defined by finite racks. These invariants reduce to the usual quandle counting invariant when the rack in question is a quandle. We are able to further enhance these counting invariants with 2-cocycles from the coloring rack's second rack cohomology satisfying a new degeneracy condition which reduces to the usual case for quandles.
We compute many dimensions of spaces of finite type invariants of virtual knots (of several kinds) and the dimensions of the corresponding spaces of "weight systems", finding everything to be in agreement with the conjecture that "every weight system integrates".
A birack is an algebraic structure with axioms encoding the blackboard-framed Reidemeister moves, incorporating quandles, racks, strong biquandles and semiquandles as special cases. In this paper we extend the counting invariant for finite racks to the case of finite biracks. We introduce a family of biracks generalizing Alexander quandles, $(t,s)$-racks, Alexander biquandles and Silver-Williams switches, known as $(τ,σ,ρ)$-biracks. We consider enhancements of the counting invariant using writhe vectors, image subbiracks, and birack polynomials.
In the present paper, we study the finite type invariants of Gauss words. In the Polyak algebra techniques, we reduce the determination of the group structure to transformation of a matrix into its Smith normal form and we give the simplified form of a universal finite type invariant by means of the isomorphism of this transformation. The advantage of this process is that we can implement it as a computer program. We obtain the universal finite type invariant of degree 4, 5, and 6 explicitly. Moreover, as an application, we give the complete classification of Gauss words of rank 4 and the part…
We enhance the biquandle counting invariant using elements of truncated biquandle-labeled Polyak algebras. These finite type enhancements reduce to the finite type enhancements defined by Goussarov, Polyak and Viro for the trivial biquandle of one element and determine (but are not determined by) the biquandle counting invariant for general biquandles. Unlike the unlabeled case, biquandle labeled finite type invariants of degree 1 are nontrivial and are related to biquandle cocycle invariants.
We define an invariant of tangles and framed tangles given a finite crossed module and a pair of functions, called a Reidemeister pair, satisfying natural properties. We give several examples of Reidemeister pairs derived from racks, quandles, rack and quandle cocycles, 2-crossed modules and braided crossed modules. We prove that our construction includes all rack and quandle cohomology (framed) link invariants, as well as the Eisermann invariant of knots, for which we also find a lifting by using braided crossed modules.
