  We denote by Q_F the family of the Alexander quandle structures supported by
finite fields. For every k-component oriented link L, every partition P of L
into h:=|P| sublinks, and every labelling z of such a partition by the natural
numbers z_1,...,z_n, the number of X-colorings of any diagram of (L,z) is a
well-defined invariant of (L,P), of the form q^(a_X(L,P,z)+1) for some natural
number a_X(L,P,z). Letting X and z vary in Q_F and among the labellings of P,
we define a derived invariant A_Q(L,P)=sup a_X(L,P,z).
  If P_M is such that |P_M|=k, we show that A_Q(L,P_M) is a lower bound for
t(L), where t(L) is the tunnel number of L. If P is a "boundary partition" of L
and g(L,P) denotes the infimum among the sums of the genera of a system of
disjoint Seifert surfaces for the L_j's, then we show that A_Q(L,P) is at most
2g(L,P)+2k-|P|-1. We set A_Q(L):=A_Q(L,P_m), where |P_m|=1. By elaborating on a
suitable version of a result by Inoue, we show that when L=K is a knot then
A_Q(K) is bounded above by A(K), where A(K) is the breadth of the Alexander
polynomial of K. However, for every g we exhibit examples of genus-g knots
having the same Alexander polynomial but different quandle invariants A_Q.
Moreover, in such examples A_Q provides sharp lower bounds for the genera of
the knots. On the other hand, A_Q(L) can give better lower bounds on the genus
than A(L), when L has at least two components.
  We show that in order to compute A_Q(L) it is enough to consider only
colorings with respect to the constant labelling z=1. In the case when L=K is a
knot, if either A_Q(K)=A(K) or A_Q(K) provides a sharp lower bound for the knot
genus, or if A_Q(K)=1, then A_Q(K) can be realized by means of the proper
subfamily of quandles X=(F_p,*), where p varies among the odd prime numbers.
