On standard forms of 1-dominations between knots with same Gromov volumes

Let k and k′ be two knots in the 3-sphere. Say k 1-dominates k′, if there is a proper degree 1 map f: E(k) → E(k′). Theorem: Suppose that any companion of k is prime. If k 1-dominates k′ with the same Gromov volume, then k′ can be obtained from k by finitely many de-satellizations. The condition of "same Gromov volume" clearly cannot be removed. We also give a new construction of 1-domination between knots with the same Gromov volume to show that the condition "any companion of k is prime" cannot be removed.