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A proof of Tait's Conjecture on alternating-achiral knots

In this paper we are interested in symmetries of alternating knots, more precisely in those related to achirality. We call the following statement Tait's Conjecture on alternating -achiral knots: Let K be an alternating -achiral knot. Then there exists a minimal projection Π of K in S^2 \subset S^3 and an involution ϕ:S^3\toS^3 such that: 1) ϕ reverses the orientation of $S^3$; 2) ϕ(S^2) = S^2; 3) ϕ (Π) = Π; 4) ϕ has two fixed points on Π and hence reverses the orientation of K. The purpose of this paper is to prove this statement.