Mutation and volumes of knots in S 3
A mutation is an operation which associates a new knot K' to a knot \(K\subset S\) 3 as follows: Isotop the knot K so that it intersects transversely the unit 2-sphere S \(2\subset {\mathbb{R}}\) \(3=S\) 3-\(\infty\) in exactly the 4 points \((\pm 1,0,0)\), \((0,\pm 1,0)\); then K' is the union of K-B 3 and of R(K\(\cap B\) 3), where B 3 is the unit ball in \({\mathbb{R}}^ 3 \)and where R is the 180 o-rotation around the x-axis. The knots K and K' may be distinct, but it has been observed for a long time that many of their invariants are equal. For instance, they cannot be distinguished by their signature, their Alexander polynomial, their Jones polynomial, or the new 2-variable polynomial invariant generalizing these. The author goes one step further in this direction by proving that their complements S 3-K and S 3-K' have the same Gromov simplicial volume. In particular, if S 3-K admits a hypebolic metric (of finite volume) then S 3-K' admits a hyperbolic metric of the same volume. The proof is based on a cut-and- paste agument in hyperbolic geometry.
scientific article; zbMATH DE number 4032621
scientific article