4-moves and the Dabkowski-Sahi invariant for knots (Q2872790)

An \(n\)-move on a knot diagram consists of inserting \(n\) positive half-twists in sequence in two parallel strands. These were first studied by \textit{S. Kinoshita} [Osaka Math. J. 9, 61--66 (1957; Zbl 0080.16903)]. For any \(n\), we call an \(n\)-move an unknotting operation if any knot can be transformed into the unknot or the unlink via \(n\)-moves. \textit{M. K. Dabkowski} and \textit{J. H. Przytycki} showed in [Geom. Topol. 6, 355--360 (2002; Zbl 1009.57013)] that \(1\)-moves and \(2\)--moves are unknotting operations, wherease \(n\)-moves are not unknotting operations for \(n=3\) or \(n\geq 5\). The same question about \(4\)-moves remains open and is Problem 1.59(3)(a) on the Kirby list of problems. To study this problem, \textit{M. K. Dabkowski} and \textit{R. K. Sahi} introduced the \(4\)-move invariant in [J. Knot Theory Ramifications 16, No. 10, 1261--1282 (2007; Zbl 1149.57005)], which is a quotient of the knot group that is invariant under \(4\)-moves. In the paper under review, the authors study the \(4\)-move invariant and develop techniques for computing it. For several classes of knots, they show that it is equal to the invariant for the unknot and thus, in these cases the invariant cannot detect a counterexample to the \(4\)-move conjecture. The main tool is a new \(4\)-move invariant for a knot, defined as a quotient of the previous one. The authors show that new invariant is equally good when looking for a counterexample to the 4-move conjecture, i.e. the previous \(4\)-move invariant has trivial value if and only if the new one does. Additionally the new invariant is more suitable for algorithmic methods of computation.