Some families of minimal elements for a partial ordering on prime knots (Q342975)

The character variety of a finitely presented group, and in particular of a \(3\)-manifold group, into \(\mathrm{SL}_2(\mathbb{C})\) has been intensively studied as it reflects geometric and topological properties of the \(3\)-manifold. In particular, this algebraic set has been used to study knots \(K\subset S^3\). For example, the number of irreducible components of the character variety \(X(K)\) of \(K\) determines minimal elements for a partial order on the set of prime knots in \(S^3\). More precisely, hyperbolic knots in \(S^3\) such that \(X(K)\) has only two irreducible components are minimal elements for that partial order.NEWLINENEWLINEIn this paper, the authors prove the minimality of all twist knots and certain double twist knots. Their proof relies on calculations of presentations of the character variety of the considered knots using Chebyshev polynomials and a criterion for irreducibility of a polynomial of two variables.