Knots on a positive template have a bounded number of prime factors.

Sullivan, Michael C.

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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 a prime alternating $-$ achiral knot. Then there exists a minimal projection $Π$ of $K$ in $S^{2} \subset S^{3}$ and an involution $ϕ : S^{3} \to S^{3}$ 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...