The stable concordance genus (Q2339290)

Let \(K\) be a knot in \(S^3\). The concordance genus of \(K\), \(g_c(K)\), is defined as the least genus of any knot in the concordance class of \(K\). It clearly satisfies \(g_4\leq g_c(K) \leq g_3(K)\), where \(g_3\), \(g_4\) denote the \(3\)-genus and \(4\)-genus, respectively. In this paper the author introduces a new invariant to study the concordance genus, called the stable concordance genus of a knot, which is defined as \(\underrightarrow{g_c(K)}:= \lim_{n \to \infty} {g_3(nK)}/{n}\), where \(nK\) denotes the connected sum of \(n\) copies of \(K\). This is is defined in a similar way as the stable 4-genus \(\underrightarrow{g_4(K)}\), defined by \textit{C. Livingston} [Algebr. Geom. Topol. 10, No. 4, 2191--2202 (2010; Zbl 1213.57015)]. Clearly \(\underrightarrow{g_c(K)}\leq g_c(K)\), but they may be different. It follows from the subaditivity of \(g_c\) that \(\underrightarrow{g_c}\) is multiplicative, that is, \(\underrightarrow{g_c(mK)}=m\underrightarrow{g_c(K)}\). Any two concordant knots have the same stable concordance genus, so \(\underrightarrow{g_c}\) can be considered as a function on the concordance group \(\mathcal{C}\). The invariant can be extended, by multiplicativity, to \(\mathcal{C}_\mathbb{Q}=\mathcal{C}\otimes \mathbb{Q}\). It follows that \(\underrightarrow{g_c}\) is a function which is multiplicative, subadditive and non-negative, thus it is a seminorm. It is shown that \(\frac{1}{2}| \sigma (K)| \leq \underrightarrow{g_c(K)} \leq g_c(K)\). In particular if \(K\) is slice, then it is stably slice, that is \({\underrightarrow{g_c(K)}=0}\). In fact, any knot which has finite order in \(\mathcal{C}\) is stably slice. The concordance polynomial of a knot \(K\) is the maximal degree polynomial which divides the Alexander polynomial of all knots concordant to \(K\), it is denoted by \(\Delta_K^c(t)\). It is shown that \(\frac{1}{2}deg(\Delta_K^c) \leq g_c(K)\). These criteria are used to calculate the stable concordance genus for all prime knots with 8 crossings or less (except for \(7_7\), \(8_1\), \(8_{13}\) and \(8_{21}\)). Then using these criteria, the stable concordance genus is calculated for knots of the form \(xT_{2,n}+yT_{2.m}\), where \(T_{2,p}\) denotes the \((2,p)\) torus knot. Namely, \(\underrightarrow{g_c}(xT_{2,n}+yT_{2.m})= \frac{| n| -1}{2}| x | +\frac{| m| - 1}{2}| y |\), where \(n,m\in \mathbb{Z}\), \(n<m\), \(kn\not= m\) for any \(k\in \mathbb{Z}\), and \(x,y \in \mathbb{Q}\). It is also proved that \(\underrightarrow{g_4(K)}\) and \(\underrightarrow{g_c(K)}\) can be different, for it is shown that for any \(j,k\in \mathbb{Q}\), for which \(1\leq j \leq k\), there is some \(K\in \mathcal{C}_\mathbb{Q}\), for which \(\underrightarrow{g_4(K)}= j\), \(\underrightarrow{g_c(K)} = k\).