An integral invariant from the knot group

Abstract

For a knot $K$ in $S^{3}$, J. Ma and R. Qiu defined an integral invariant $a(K)$ which is the minimal number of elements that generate normally the commutator subgroup of the knot group, and showed that it is a lower bound of the unknotting number. We prove that it is also a lower bound of the tunnel number. If the invariant were additive under connected sum, then we could deduce something about additivity of both the unknotting numbers and the tunnel numbers. However, we found a sequence of examples that the invariant is not additive under connected sum. Let $T(2, p)$ be the $(2, p)$-torus knot, and $K_{p, q}=T(2, p) \sharp T(2, q)$. Then we have $a(K_{p, q})=1$ if and only if $\gcd(p, q)= 1$.