An integral invariant from the knot group
The Nakanishi-index \(m(K)\) of a knot \(K\) in \(S^3\) is the minimal number of generators of the Alexander module of \(K\). The authors propose the name MQ-index for the invariant \(a(K)\) considered by Ma and Qiu, which is the minimal number of elements of the knot group of \(K\) that generate normally the commutator subgroup. Ma and Qiu showed that \(m(K)\leq a(K)\leq u(K)\), where \(u(K)\) is the unknotting number. The authors show that \(a(K)\leq \min\{r(K)-1, u(K)\} \leq \min\{t(K), u(K)\}\), where \(r(k)\) is the rank of the knot group and \(t(K)\) is the tunnel number. It is known that the tunnel number is not additive with respect to connected sums of knots, but this is not known for the unknotting number. The authors show that \(a(K)\) is not additive by proving that for the connected sum \(K_{p,q} \) of two torus knots \(T(2,p)\) and \(T(2,q)\) with \(3\leq p\leq |q|\), \,\(a(K_{p,q})=1 \) if and only if \(p\) and \(q\) are relatively prime, and this is true if and only if \(m(K_{p,q})=1\).
scientific article; zbMATH DE number 5848637
scientific article