Ribbon crossing numbers, crossing numbers, and Alexander polynomials
Ribbon \(2\)-knotted objects are locally flat embeddings of surfaces in \(4\)-space which bound immersed \(3\)-manifolds with only ribbon singularities. They appear as topological realizations of welded knotted objects, which is a natural quotient of virtual knot theory. Let \(K^2\) be a ribbon \(2\)-knot. The ribbon crossing number, denoted here by \(r-cr(K^2)\) is a numerical invariant of the ribbon \(2\)-knot \(K^2\). It is known that the degree (the difference of between the maximal degree and the minimal degree) of the Alexander polynomial of \(K^2\) is less than or equal to \(r-cr(K^2)\). In the paper under review the author shows that \(r-cr(K^2)\) is estimated by coefficients of the Alexander polynomial of \(K^2\). As an application, the author obtains an estimation of the crossing number, denoted by \(cr(k^1)\), for a classical knot \(k^1\).
scientific article; zbMATH DE number 6924615