Related papers: Implementation of the Habegger--Lin decision algor…
Given an $m$-component link $L$ in $S^3$ ($m \ge 2$), we construct a family of links which are link homotopic, but not link isotopic, to $L$. Every proper sublink of such a link is link isotopic to the corresponding sublink of $L$.…
Polyak showed that any Milnor's $\overline{\mu}$-invariant of length 3 can be represented as a combination of Conway polynomials of knots obtained by certain band sum of the link components. On the other hand, Habegger and Lin showed that…
A categorification of a polynomial link invariant is an homological invariant which contains the polynomial one as its graded Euler characteristic. This field has been initiated by Khovanov categorification of the Jones polynomial. Later,…
We define a triply-graded invariant of links in a genus g handlebody, generalizing the colored HOMFLYPT (co)homology of links in the 3-ball. Our main tools are the description of these links in terms of a subgroup of the classical braid…
Tied links and the tied braid monoid were introduced recently by the authors and used to define new invariants for classical links. Here, we give a version purely algebraic-combinatoric of tied links. With this new version we prove that the…
The A-B slice problem is a reformulation of the topological 4-dimensional surgery conjecture in terms of decompositions of the 4-ball and link homotopy. We show that link groups, a recently developed invariant of 4-manifolds, provide an…
For an $n$-component link $L$, the Milnor's isotopy invariant is defined for each multi-index $I=i_1i_2...i_m (i_j\in\n)$. Here $m$ is called the length. Let $r(I)$ denote the maximam number of times that any index appears. It is known that…
Link-homotopy and self Delta-equivalence are equivalence relations on links. It was shown by J. Milnor (resp. the last author) that Milnor invariants determine whether or not a link is link-homotopic (resp. self Delta-equivalent) to a…
In this article we study the Heegaard Floer link homology of $(n, n)$-torus links. The Alexander multigradings which support non-trivial homology form a string of $n-1$ unit hypercubes in $\mathbb{R}^{n}$, and we compute the ranks and…
The notion of a Bing cell is introduced, and it is used to define invariants, link groups, of 4-manifolds. Bing cells combine some features of both surfaces and 4-dimensional handlebodies, and the link group \lambda(M) measures certain…
We define an integer valued invariant for two-component links in S^3 by counting projective SU(2) representations of the link group having non-trivial second Stiefel-Whitney class. We show that our invariant is, up to sign, the linking…
We affirmatively address the question of whether the proposed link homotopy invariant $\omega$ of Li is well-defined. It is also shown that if one wishes to adapt the homotopy invariant $\tau$ of Schneiderman-Teichner to a link homotopy…
This paper describes an approach to computer aided calculations in the cohomology of arithmetic groups. It complements existing literature on the topic by emphasizing homotopies and perturbation techniques, rather than cellular subdivision,…
We define a notion of finite type invariants for links with a fixed linking matrix. We show that Milnor's triple link homotopy invariant is a finite type invariant, of type 1, in this sense. We also generalize the approach to Milnor's…
Motivated by recent best case analyses for some sorting algorithms and based on the type of complexity we partition the algorithms into two classes: homogeneous and non homogeneous algorithms. Although both classes contain algorithms with…
In this paper, we use a partition of the links of a network in order to uncover its community structure. This approach allows for communities to overlap at nodes, so that nodes may be in more than one community. We do this by making a node…
Identification of communities in complex networks has become an effective means to analysis of complex systems. It has broad applications in diverse areas such as social science, engineering, biology and medicine. Finding communities of…
In this paper, we study the equivariant homotopy type of a connected sum of linear actions on complex projective planes defined by Hambleton and Tanase. These actions are constructed for cyclic groups of odd order. We construct cellular…
We use Kirk's invariant of link maps $S^2\sqcup S^2\to S^4$ and its variations due to Koschorke and Kirk-Livingston to deduce results about classical links. Namely, we give a new proof of the Nakanishi-Ohyama classification of two-component…
We study how to detect groups in a complex network each of which consists of component nodes sharing a similar connection pattern. Based on the mixture models and the exploratory analysis set up by Newman and Leicht (Newman and Leicht 2007…