Grids as unknotting operations
A grid local move of type \(a\times b\) is a local simultaneous crossing change on a \(a\times b\) grid. A \(1\times 1\) grid move is a crossing change and a \(2\times 2\) grid move is a \(\#\)-unknotting operation introduced by \textit{H. Murakami} [Math. Ann. 270, 35-45 (1985; Zbl 0535.57005)]. The main theorem says that a glid local move of type \(a\times b\) is an unknotting operation for \(a= 3,4\) and any \(b\geq 1\) modulo that fact a move of type \(4\times 2b\) has to preserve the Arf invariants. Namely, (1) a \(3\times b\) grid move with \(b\geq 1\) is an unknotting operation, (2) a \(4\times b\) grid move with \(b\) odd is an unknotting operation, and (3) if \(b\) is an even positive integer, then every knot is transformed into either the unknot or a trefoil knot by \(4\times b\) grid moves according as the Arf invariant of \(K\) is \(0\) or \(1\). The author also gives an estimate of the corresponding unknotting numbers in terms of the local signatures.NEWLINENEWLINEFor the entire collection see [Zbl 0959.00034].
scientific article; zbMATH DE number 1619399