Narrow Results

We adopt the notion of combability of groups defined in [12]. An example is given of a (bi-)combable group which is not residually finite. Two of the eight 3-dimensional geometries, and ℍ²×ℝ, are quasiisometric. Seifert fibred manifolds over hyperbolic orbifolds have bicombable fundamental groups. Every combable group satisfies an exponential isoperimetric inequality.

We describe an algorithm to compute the Kuperberg invariant of a combed or framed 3–manifold, starting from a presentation of such a manifold in terms of branched standard spines.

We describe an algorithm to compute the Kuperberg invariant of a combed or framed 3–manifold, starting from a presentation of such a manifold in terms of branched standard spines.

Tied links in $S^3$ were introduced by Aicardi and Juyumaya as standard links in $S^3$ equipped with some non-embedded arcs, called ties, joining some components of the link. Tied links in the Solid Torus were then naturally generalized by Flores. In this paper, we study this new class of links in other topological settings. More precisely, we study tied links in the lens spaces $L(p,1)$, in handlebodies of genus $g$, and in the complement of the $g$-component unlink. We introduce the tied braid monoids $T{M}_{g,n}$ by combining the algebraic mixed braid groups defined by Lambropoulou and the tied braid monoid, and we formulate and prove analogues of the Alexander and the Markov theorems for tied links in the 3-manifolds mentioned above. We also present an $L$-move braid equivalence for tied braids and we discuss further research related to tied links in knot complements and c.c.o. 3-manifolds. The theory of tied links has potential use in some aspects of molecular biology.