-knots via the mapping class group of the twice punctured torus.
1-slim triangles and uniform hyperbolicity for arc graphs and curve graphs
We describe unicorn paths in the arc graph and show that they form 1-slim triangles and are invariant under taking subpaths. We deduce that all arc graphs are 7-hyperbolic. Considering the same paths in the arc and curve graph, this also shows that all curve graphs are 17-hyperbolic, including closed surfaces.
-dimensional Euclidean manifolds represented by locally regular coloured graphs
3-coloring and other elementary invariants of knots
3-manifold invariants and periodicity of homology spheres.
3-manifold spines and bijoins.
We describe a combinatorial algorithm for constructing all orientable 3-manifolds with a given standard bidimensional spine by making use of the idea of bijoin (Bandieri and Gagliardi (1982), Graselli (1985)) over a suitable pseudosimplicial triangulation of the spine.
3-manifolds with(out) metrics of nonpositive curvature.
4--manifolds as covers of the 4--sphere branched over non-singular surfaces.
ℓ²-homology and planar graphs
In his 1930 paper, Kuratowski proves that a finite graph Γ is planar if and only if it does not contain a subgraph that is homeomorphic to K₅, the complete graph on five vertices, or , the complete bipartite graph on six vertices. This result is also attributed to Pontryagin. In this paper we present an ℓ²-homological method for detecting non-planar graphs. More specifically, we view a graph Γ as the nerve of a related Coxeter system and construct the associated Davis complex, . We then use a...
...-reducing Dehn surgeries and 1-bridge knots.
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-curves inducing two different knots with the same -fold branched covering spaces
For a knot with a strong inversion induced by an unknotting tunnel, we have a double covering projection branched over a trivial knot , where is the axis of . Then a set is called a -curve. We construct -curves and the cyclic branched coverings over -curves, having two non-isotopic Heegaard decompositions which are one stable equivalent.
-chamber systems, coloured graphs and orbifolds.
Алгебраические уравнения с непрерывными коэффициентами и некоторые вопросы алгебраической теории кос
Бесконечные соизмеримые гиперболические группы билипшицево эквиваленты.
Вложение открытых многообразий в компактные.
Геометрическая характеризация свободных конструкций проконечных групп.
Геометрическое разложение пространственных клейновых групп и фундаментальные группы 3-многообразий.
Гиперболические объёмы многообразий Фибоначчи.