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On malnormal peripheral subgroups of the fundamental group of a 3 -manifold

Pierre de la Harpe, Claude Weber (2014)

Confluentes Mathematici

Let K be a non-trivial knot in the 3 -sphere, E K its exterior, G K = π 1 ( E K ) its group, and P K = π 1 ( ∂ E K ) ⊂ G K its peripheral subgroup. We show that P K is malnormal in G K , namely that g P K g - 1 ∩ P K = { e } for any g ∈ G K with g ∉ P K , unless K is in one of the following three classes: torus knots, cable knots, and composite knots; these are exactly the classes for which there exist annuli in E K attached to T K which are not boundary parallel (Theorem 1 and Corollary 2). More generally, we characterise malnormal peripheral subgroups in the fundamental group of a...

On manifold spines and cyclic presentations of groups

Alberto Cavicchioli, Friedrich Hegenbarth, Dušan Repovš (1998)

Banach Center Publications

This is a survey of results and open problems on compact 3-manifolds which admit spines corresponding to cyclic presentations of groups. We also discuss questions concerning spines of knot manifolds and regular neighborhoods of homotopically PL embedded compacta in 3-manifolds.

On numerical invariants for knots in the solid torus

Khaled Bataineh (2015)

Open Mathematics

We define some new numerical invariants for knots with zero winding number in the solid torus. These invariants explore some geometric features of knots embedded in the solid torus. We characterize these invariants and interpret them on the level of the knot projection. We also find some relations among some of these invariants. Moreover, we give lower bounds for some of these invariants using Vassiliev invariants of type one. We connect our invariants to the bridge number in the solid torus. We...

On polynomials and surfaces of variously positive links

Alexander Stoimenow (2005)

Journal of the European Mathematical Society

It is known that the minimal degree of the Jones polynomial of a positive knot is equal to its genus, and the minimal coefficient is 1, with a similar relation for links. We extend this result to almost positive links and partly identify the next three coefficients for special types of positive links. We also give counterexamples to the Jones polynomial-ribbon genus conjectures for a quasipositive knot. Then we show that the Alexander polynomial completely detects the minimal genus and fiber property...

On real algebraic links in S 3

R. Benedetti, M. Shiota (1998)

Bollettino dell'Unione Matematica Italiana

Viene presentata una costruzione che, dato un arbitrario nodo L ⊆ S 3 , produce allo stesso tempo: 1) un'applicazione polinomiale f : R 4 , 0 → R 2 , 0 con singolarità (debolmente) isolata in 0 e L come tipo di nodo della singolarità; 2) una risoluzione delle singolarità di f nel senso di Hironaka. Specializzando la costruzione ai nodi fibrati otteniamo una versione debole (a meno di scoppiementi e nella categoria analitica reale) di un reciproco per il teorema di fibrazione di Milnor.

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