In this paper, we study distance one surgeries between lens spaces L(p, 1) with p ≥ 5 prime and lens spaces L(n, 1) for $$n \in \mathbb{Z}$$ and band surgeries from T (2, p) to T (2, n). In particular, we prove that L(n, 1) is obtained by a distance one surgery from L(5, 1) only if n=±1, 4, ±5, 6 or ±9, and L(n, 1) is obtained by a distance one surgery from L(7, 1) if and only if n=±1, 3, 6, 7, 8 or 11.

Solving tangle equations is deeply connected with studying enzyme action on DNA. The main goal of this paper is to solve the system of tangle equations [Formula: see text] and [Formula: see text], where [Formula: see text] and [Formula: see text] are rational tangles, and [Formula: see text] is a 2-bridge link, for [Formula: see text], with [Formula: see text] and [Formula: see text] nontrivial. We solve this system of equations under the assumption [Formula: see text], the double branched cover of [Formula: see text], is not hyperbolic, i.e. [Formula: see text] is not [Formula: see text]-hyperbolic. Besides, we also deal with tangle equations involving 2-bridge links only under the assumption [Formula: see text] is an algebraic tangle.

Solving tangle equations is deeply connected with studying enzyme action on DNA. The main goal of this paper is to solve the system of tangle equations, where X 1 and X 2 are rational tangles, and b i is a 2-bridge link, for i = 1, 2, 3, with b 2 and b 3 nontrivial. We solve this system of equations under the assumption O, the double branched cover of O, is not hyperbolic, i.e.O is not π -hyperbolic. Besides, we also deal with tangle equations involving 2-bridge links only under the assumption O is an algebraic tangle.

In this paper, we study distance one surgeries between lens spaces L(p, 1) with p ≥ 5 prime and lens spaces L(n, 1) for n ∈ Z and band surgeries from T (2, p) to T (2, n). In particular, we prove that L(n, 1) is obtained by a distance one surgery from L(5, 1) only if n = ±1, 4, ±5, 6 or ±9, and L(n, 1) is obtained by a distance one surgery from L(7, 1) if and only if n = ±1,