Geodesic
Meridional rank and bridge number of knotted 2-spheres
Canadian journal of mathematics, Tome 77 (2025) no. 1, pp. 282-299

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DOI

The meridional rank conjecture asks whether the bridge number of a knot in $S^3$ is equal to the minimal number of meridians needed to generate the fundamental group of its complement. In this paper, we investigate the analogous conjecture for knotted spheres in $S^4$. Towards this end, we give a construction to produce classical knots with quotients sending meridians to elements of any finite order in Coxeter groups and alternating groups, which detect their meridional ranks. We establish the equality of bridge number and meridional rank for these knots and knotted spheres obtained from them by twist-spinning. On the other hand, we show that the meridional rank of knotted spheres is not additive under connected sum, so that either bridge number also collapses, or meridional rank is not equal to bridge number for knotted spheres.
DOI : 10.4153/S0008414X23000883
Mots-clés : Meridional rank, bridge number, 2-knots, knotted surfaces, Coxeter groups 
Joseph, Jason; Pongtanapaisan, Puttipong. Meridional rank and bridge number of knotted 2-spheres. Canadian journal of mathematics, Tome 77 (2025) no. 1, pp. 282-299. doi: 10.4153/S0008414X23000883
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