On the crossing number of 2-bridge knot and the canonical genus of its Whitehead double (Q854534)

A Seifert surface of a knot \(K\) in the 3-sphere \(\mathbb{S}^3\) is a compact connected orientable surface \(S\) in \(\mathbb{S}^3\) such that the boundary of \(S\) is \(K\). The minimal genus among all Seifert surfaces of \(K\) is called the genus of \(K\), denoted by \(g(K)\). A Seifert surface of \(K\) is said to be canonical if it is obtained from a diagram of \(K\) by applying Seifert's algorithm. Then the minimal genus among all canonical Seifert surfaces of \(K\) is called the canonical genus of \(K\), denoted by \(g_c(K)\). It follows that \(g(K)\leq g_c(K)\). It is known for example that a twisted Whitehead double of the trefoil has the canonical genus at least three but its genus is one. The author proves that the crossing number of a 2-bridge knot coincides with the canonical genus of its Whitehead double.