On wrapping number, adequacy and the crossing number of satellite knots

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In this work we establish the tightest lower bound up-to-date for the minimal crossing number of a satellite knot based on the minimal crossing number of the companion used to build the satellite. If M is the wrapping number of the pattern knot, we essentially show that c(Sat(P,C))>M²/2c(C). The existence of this bound will be proven when the companion knot is adequate, and it will be further tuned in the case of the companion being alternating.