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10.26 (1986)

Partially Solved

a) Does there exist an algorithm which decides, for given elements $a, b$ and an automorphism $\phi$ of a free group, whether the equation $a x^\phi = xb$ is soluble in this group? This question seems to be useful for solving the problem of equivalence of two knots.
b) Does there exist an algorithm which decides whether the equation of the form $w(x_{i_1}^{\varphi_1}, \dots, x_{i_n}^{\varphi_n}) = 1$ is soluble in a free group where $\varphi_1, \dots, \varphi_n$ are automorphisms of this group?

Contributor: G. S. Makanin

Progress

a) Yes, it does (O. Bogopolski, A. Martino, O. Maslakova, E. Ventura, Bull. London Math. Soc., 38, no. 5 (2006), 787–794); further generalizations and applications are in (O. Bogopolski, A. Martino, E. Ventura, Trans. Amer. Math. Soc., 362 (2010), 2003–2036).

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