33 research outputs found
Graph-Links
The present paper is a review of the current state of Graph-Link Theory
(graph-links are also closely related to homotopy classes of looped
interlacement graphs), dealing with a generalisation of knots obtained by
translating the Reidemeister moves for links into the language of intersection
graphs of chord diagrams. In this paper we show how some methods of classical
and virtual knot theory can be translated into the language of abstract graphs,
and some theorems can be reproved and generalised to this graphical setting. We
construct various invariants, prove certain minimality theorems and construct
functorial mappings for graph-knots and graph-links. In this paper, we first
show non-equivalence of some graph-links to virtual links.Comment: 32 pages, 21 figure
A new proof that alternating links are non-trivial
We use a simple geometric argument and small cancellation properties of link
groups to prove that alternating links are non-trivial. This proof uses only
classic results in topology and combinatorial group theory.Comment: Minor changes. To appear in Fundamenta Mathematica
Binary matroids and local complementation
We introduce a binary matroid M(IAS(G)) associated with a looped simple graph
G. M(IAS(G)) classifies G up to local equivalence, and determines the
delta-matroid and isotropic system associated with G. Moreover, a parametrized
form of its Tutte polynomial yields the interlace polynomials of G.Comment: This article supersedes arXiv:1301.0293. v2: 26 pages, 2 figures. v3
- v5: 31 pages, 2 figures v6: Final prepublication versio