The number of Reidemeister moves needed for unknotting (Q2701705)

It is proved that any unknotted diagram with \(n\) crossings can be transformed into the trivial diagram using at most \(2^{c_{1}n}\) Reidemeister moves for a positive constant \(c_1\). NEWLINENEWLINENEWLINEThe proof proceeds as follows. NEWLINENEWLINENEWLINEThe goal is to prove that for any compact oriented triangulated three-manifold with boundary \((M,\partial{M})\) with \(t\) tetrahedra and an unknot \(K\) in the one-skeleton of the interior of \(M\), \(K\) can be isotoped to a triangle (a face of a tetrahedron) using at most \(2^{c_{2}t}\) elementary moves for a constant \(c_2\). Here an elementary move, roughly speaking, changes an edge in a triangle into the other two in the same triangle. NEWLINENEWLINENEWLINETo prove the above estimation, one uses the normal surface theory [\textit{W. Haken}, Acta Math. 105, 245--375 (1961; Zbl 0100.19402); \textit{W. Jaco} and \textit{J. L. Tollefson}, Ill. J. Math. 39, No. 3, 358--406 (1995; Zbl 0858.57018)]. First one removes from \(M\) the regular neighborhood \(R_{K}\) of \(K\), giving \(M_{K}=M\setminus{R_{K}}\) with new boundary \(\partial{M_{K}}\), called the peripheral torus. Since \(K\) is unknotted there exists an essential normal disk \(S\) in \(M_{K}\). Then by \textit{J. Hass, J. C. Lagarias} and \textit{N. Pippenger} [The computational complexity of knot and link problems, J. ACM 46, No. 2, 185--211 (1999; Zbl 1065.68667)], one may assume that \(S\) contains at most \(2^{8t+6}\) triangles. It can be proved that the boundary curve \(K_{2}=\partial{S}\) can be moved to a single triangle in \(S\) across triangles, involving \(2^{8t+7}\) elementary moves. NEWLINENEWLINENEWLINEOn the other hand, one can construct a longitude \(K_{1}\) in the one-skeleton of the peripheral torus such that it has at most \(O(2^{c_{3}t})\) edges and can be isotoped to \(K\) using at most \(O(2^{c_{3}t})\) elementary moves. Since \(K_{1}\) can be transformed to \(K_{2}\) on \(\partial{R_{K}}\) using at most \(O(2^{c_{4}t})\) elementary moves, the estimation follows. NEWLINENEWLINENEWLINEAs a corollary to the main result one can see that any unknotted diagram with \(n\) crossings can be transformed into a trivial knot diagram through a sequence of diagrams each of which has at most \(2^{c_{1}n}\) crossings. NEWLINENEWLINENEWLINESimilar estimation based on the normal surface theory is also obtained by \textit{S. Galatolo} [On a problem of effective knot theory, Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl. 9, No. 4, 299--306 (1999; Zbl 1001.57007)].