Abstract: Let be a knot in the three dimensional sphere 3. Denote by a cyclic -fold covering of 3 branched over . The results presented in this lecture are motivated by the classical theorem of Spanish mathematician Antonio Plans (1953) which told us that 1( ,ℤ)= ⊕ if is odd and Ker( 1( ,ℤ)→ 1( 2,ℤ))= ⊕ if is even, for a suitable Abelian group . Our goal is to give a constructive proof of Plans theorem for the first homology group of cyclic coverings of knots as well as for the critical group of cyclic branched coverings of graphs. This is joint work with Lilia Grunwald, Young Soo Kwon, Ilya Mednykh and Galina Sokolova.

Cite: Mednykh A.D.
Plans theorem for knots and graphs
The First International On-line Knot Theory Congress 01-06 Feb 2025