Yevamos 44a - Oleh L'chalitzha Oleh L'yibum

The gemara says a rule that all who are bound for yibum are bound for chalitzah, but all who are not bound for yibum are not bound for chalitzah. Tosafos assumes that there is also a rule that all who are not oleh l'chalitzah are not oleh l'yibum. Although the gemara never made such a statement, if we follow the rules of logic it is nonetheless true. The rules of logic are that the contra positive is always true, namely "If A then B = If not B then Not A". Since All who are oleh l'yibum are oleh l'chalitzah, therefore All who are not oleh l'chalitzah are not oleh l'yibum. This is why Tosafos assumption is correct (obviously this only works since the definition of "If A then B" is not that the only way to achieve B is A, rather that if one does A they will definitely achieve B so it is impossible to do A and not achieve B, therefore if one did not achieve B that is bec. they did not do A).

On a more lomdush note:

Tosafos says that although if one is not eligible for chalitzah they are not eligible for yibum, a deaf/mute is eligible for chalitzah just impossible for him to perform it, and he therefore is eligible for chalitzah. R' Akiva Eiger (Mishnah Sanhedrin chapt. 2 os 7) asks that a king is considered not eligible for chalitzah since it is against the kavod hamelech for her to spit in front of him, and she is therefore exempt even from yibum. Why it this so, it should be similar to a deaf/mute who is eligible just unable to perform? R' Elchonon (5:5) suggest that we only consider the situation eligible but impossible - 'pumei ka'iv lei" when it could technically change, but an issur to spit in front of a king will always be there and is considered not eligible. This answer is difficult bec. the point is that the inability to perform chalitzah is an external factor both by the deaf/mute and by the king.

R' Moshe (Igros E.H. 1:158) gives a very clear answer, he assumes that we look at it from the perspective of the mitzvah. So, a deaf/mute has a mitzvah of chalitzah just that he is unable to perform it, so he is still obligated in the mitzvah of yibum. But a king has not mitzvah of chalitzah since it is assur to perform it, therefore he is not eligible for yibum and since there is no mitzvah of yibum or chalitzah, there is no zika.