A conjecture by Thom asserts that algebraic curves in \(\mathbb C\mathbb P^2\) are genus-minimizing in their homology class (see [P. B. Kronheimer and T. S. Mrowka, Math. Res. Lett. 1, No. 6, 797–808 (1994; Zbl 0851.57023)]) or, more generally, that no smooth embedded surface in \(\mathbb C\mathbb P^2\) has smaller genus than an algebraic curve homologous to it. On the other hand, there are topological locally flat surfaces with genus strictly smaller than all algebraic curves homologous to them (see papers by L. Rudolph [Comment. Math. Helv. 59, 592–599 (1984; Zbl 0575.57003)], and by R. Lee and D. M. Wilczyński [Am. J. Math. 119, No. 5, 1119–1137 (1997; Zbl 0886.57027)]).
The knot theoretic version of the Thom conjecture asserts that the smooth slice genus of a positive braid knot coincides with its ordinary genus \(g\) (see [L. Rudolph, Bull. Am. Math. Soc., New Ser. 29, No. 1, 51–59 (1993; Zbl 0789.57004)]). Less is known about the topological locally flat slice genus \(g_4\) of positive braid knots, or even of torus knots. In the present paper it is shown that, for torus knots \(K\), \(g_4(K) \leq {6 \over 7} \, g(K)\) (excluding some small cases). As the authors note, this result is sharp since for the torus knot \(K = T(3,8)\), \(g_4(K) = 6\) and \(g(K) = 7\). A larger genus defect is attained for torus knots \(T(p,q)\) with large parameters \(p\) and \(q\) for which \({2 \over pq} \; g(T(p,q) = {2 \over pq} \; {1 \over 2}(p-1)(q-1)\) converges to 1 whereas, as the authors show, the limit of \({2 \over pq} \; g_4(T(p,q)\) is less than \({3 \over 4}\).