string links
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2021 ◽  
pp. 1-23
Author(s):  
Haruko A. Miyazawa ◽  
Kodai Wada ◽  
Akira Yasuhara
Keyword(s):  

2020 ◽  
Vol 29 (01) ◽  
pp. 2050002
Author(s):  
R. Komendarczyk ◽  
A. Michaelides

This paper concerns the tree invariants of string links, introduced by Kravchenko and Polyak, which are closely related to the classical Milnor linking numbers also known as [Formula: see text]-invariants. We prove that, analogously as for [Formula: see text]-invariants, certain residue classes of tree invariants yield link-homotopy invariants of closed links. The proof is arrow diagramatic and provides a more geometric insight into the indeterminacy through certain tree stacking operations. Further, we show that the indeterminacy of tree invariants is consistent with the original Milnor’s indeterminacy. For practical purposes, we also provide a recursive procedure for computing arrow polynomials of tree invariants.


2019 ◽  
Vol 28 (12) ◽  
pp. 1950072
Author(s):  
Juliana Roberta Theodoro De Lima

In 1947, in the paper “Theory of Braids,” Artin raised the question of whether isotopy and homotopy of braids on the disk coincide. Twenty seven years later, Goldsmith answered his question: she proved that in fact the group structures are different, exhibiting a group presentation and showing that the homotopy braid group on the disk is a proper quotient of the Artin braid group on the disk [Formula: see text], denoted by [Formula: see text]. In this paper, we extend Goldsmith’s answer to Artin for closed, connected and orientable surfaces different from the sphere. More specifically, we define the notion of homotopy generalized string links on surfaces, which form a group which is a proper quotient of the braid group on a surface [Formula: see text], denoting it by [Formula: see text]. We then give a presentation of the group [Formula: see text] and find that the Goldsmith presentation is a particular case of our main result, when we consider the surface [Formula: see text] to be the disk. We close with a brief discussion surrounding the importance of having such a fixed construction available in the literature.


2019 ◽  
Vol 28 (03) ◽  
pp. 1950022
Author(s):  
Zhiyun Cheng ◽  
Hongzhu Gao ◽  
Mengjian Xu

In this paper, we define transcendental polynomial invariants for two-component virtual string links. One of these invariants is a strictly refinement of the linking polynomial in [M. Xu and H. Gao, Linking polynomials of virtual string links, Sci. China Math. 61(7) (2018) 1287–1302]. It is a homotopy invariant and can distinguish some virtual string links from their mirror images. We also define a transcendental polynomial invariant for two-component flat virtual string links. These invariants can be used to study the periodicity and linking crossing number of virtual string links.


2018 ◽  
Vol 27 (13) ◽  
pp. 1842012
Author(s):  
Sergey A. Melikhov

We use Kirk’s invariant of link maps [Formula: see text] 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 links in [Formula: see text] up to [Formula: see text]-link homotopy. We also prove its version for string links, which is due (in a slightly different form) to Fleming–Yasuhara. The proofs do not use Clasper Theory.


2018 ◽  
Vol 30 (5) ◽  
pp. 1209-1235
Author(s):  
Paul Arnaud Songhafouo Tsopméné ◽  
Victor Turchin

AbstractArone and the second author showed that when the dimensions are in the stable range, the rational homology and homotopy of the high-dimensional analogues of spaces of long knots can be calculated as the homology of a direct sum of finite graph-complexes that they described explicitly. They also showed that these homology and homotopy groups can be interpreted as the higher-order Hochschild homology, also called Hochschild–Pirashvili homology. In this paper, we generalize all these results to high-dimensional analogues of spaces of string links. The methods of our paper are applicable in the range when the ambient dimension is at least twice the maximal dimension of a link component plus two, which in particular guarantees that the spaces under study are connected. However, we conjecture that our homotopy graph-complex computes the rational homotopy groups of link spaces always when the codimension is greater than two, i.e. always when the Goodwillie–Weiss calculus is applicable. Using Haefliger’s approach to calculate the groups of isotopy classes of higher-dimensional links, we confirm our conjecture at the level of {\pi_{0}}.


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