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The concordance genus of a knot. II. (English) Zbl 1171.57004

This paper is a sequel to the author’s paper [Algebr. Geom. Topol. 4, 1-22 (2004; Zbl 1055.57007)]. Let \(K\) be a knot in the 3-sphere \(S^3\). Let \(g_3(K)\) be the minimum genus of an orientable embedded surface bounded by \(K\) in \(S^3\), and \(g_4(K)\) the minimum genus of an orientable embedded surface bounded by \(K\) in the 4-ball \(B^4\). We denote by \(g_c(K)\) the minimum value of \(g_3(J)\) among all knots \(J\) concordant to \(K\). This paper extends previous techniques to compute these invariants, provides interesting examples and finishes the computations for knots with 10 or fewer crossings.
Concretely, the author proved:
Proposition 1. For \(K=8_{18}\) and \(K=9_{40}\), \(g_3(K)=g_c(K)=3\) and \(g_4(K)=1\). Proposition 2. For \(K=10_{82}, g_3(K)=4\) and \(g_4(K)=1\). There are knots \(J\) with \(g_3(J)\leq 3\) that are algebraically concordant to \(K\), the first of which is \(9_{42}\), but \(K\) is not concordant to any such \(J\). In particular, \(g_{c}(K)=4\).
Further, the last section lists some related open problems.

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)
57N70 Cobordism and concordance in topological manifolds

Citations:

Zbl 1055.57007

Software:

KnotInfo
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References:

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