Related papers: Finite Type Invariants of Links with Fixed Linking…
The paper concerns the tree invariants of string links, introduced by Kravchenko and Polyak and closely related to the classical Milnor linking numbers also known as $\bar{\mu}$--invariants. We prove that, analogously as for…
This paper explicitly details the relation between $M$-matrices, nonnegative roots of nonnegative matrices, and the embedding problem for finite-state stationary Markov chains. The set of nonsingular nonnegative matrices with arbitrary…
We study a class of one-dimensional full branch maps admitting two indifferent fixed points as well as critical points and/or unbounded derivative. Under some mild assumptions we prove the existence of a unique invariant mixing absolutely…
Delta finite-type invariants are defined analogously to finite-type invariants, using delta moves instead of crossing changes. We show that they are closely related to the lower central series of the commutator subgroup of the pure braid…
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…
We define a family of formal Khovanov brackets of a colored link depending on two parameters. The isomorphism classes of these brackets are invariants of framed colored links. The Bar-Natan functors applied to these brackets produce…
This is a survey article on finite type invariants of 3-manifolds written for the Encyclopedia of Mathematical Physics to be published by Elsevier.
We show that for an $n$-component, $n$-bridge link and a positive integer $m$, the following is true: If the longitudes of $L$ lie in the $(m+2)$-th term of the lower central series of the link group then all the finite type invariants of…
We use Polyak's skein relation to give a new proof that Milnor's string link homotopy invariants are finite type invariants, and to develop a recursive relation for their associated weight systems. We show that the obstruction to the…
In this paper we give a re-normalization of the Reshetikhin-Turaev quantum invariants of links, by modified quantum dimensions. In the case of simple Lie algebras these modified quantum dimensions are proportional to the usual quantum…
We consider a class of doubly intermittent maps with critical points, unbounded derivative and regularly varying tails. Under some mild assumptions we prove the existence of a unique mixing absolutely continuous invariant measure and give…
Goussarov, Polyak, and Viro proved that finite type invariants of knots are ``finitely multi-local'', meaning that on a knot diagram, sums of quantities, defined by local information, determine the value of the knot invariant. The result…
We prove two general decomposition theorems for fixed-point invariants: one for the Lefschetz number and one for the Reidemeister trace. These theorems imply the familiar additivity results for these invariants. Moreover, the proofs of…
We introduce new topological quantum invariants of compact oriented 3-manifolds with boundary where the boundary is a disjoint union of two identical surfaces. The invariants are constructed via surgery on manifolds of the form $F \times I$…
Fixing two concordant links in $3$--space, we study the set of all embedded concordances between them, as knotted annuli in $4$--space. When regarded up to surface-concordance or link-homotopy, the set $\mathcal{C}(L)$ of concordances from…
Costantino--Geer--Patureau-Mirand proved relations between the Reshetikhin--Turaev link invariants and the re-normalized link invariants for knots. Their theorem says that residues of the re-normalized link invariants are given by the…
This is a brief introduction to link homology theories that categorify Reshetikhin--Turaev $\mathsf{SL}(N)$-quantum link invariants. A recently discovered surprising connection between finite state automata and Boolean TQFTs in dimension…
In this paper, we define and classify the sign-equivalent exchange matrices. We give a Diophantine explanation for the differences between rank 2 cluster algebras of finite type and affine type based on \cite{CL24}. We classify the positive…
Directed graphs are widely used in modelling of nonsymmetric relations in various sciences and engineering disciplines. We discuss invariants of strongly connected directed graphs - minimal number of vertices or edges necessary to remove to…
Wiring diagrams usually serve as a tool in the study of arrangements of lines and pseudolines. In this paper we go in the opposite direction, using known properties of line arrangements to motivate certain equivalence relations and actions…