Osaka Journal of Mathematics

An integral invariant from the knot group

Teruhisa Kadokami and Zhiqing Yang

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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 $\gcd(p, q)= 1$.

Article information

Osaka J. Math., Volume 47, Number 4 (2010), 965-976.

First available in Project Euclid: 20 December 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M05: Fundamental group, presentations, free differential calculus 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45} 57M27: Invariants of knots and 3-manifolds


Kadokami, Teruhisa; Yang, Zhiqing. An integral invariant from the knot group. Osaka J. Math. 47 (2010), no. 4, 965--976.

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