ALEXANDER QUANDLE LOWER BOUNDS FOR LINK GENERA
Abstract
Every finite field 𝔽q, q = pn, carries several Alexander quandle structures 𝕏 = (𝔽q, *). We denote by 
the family of these quandles, where p and n vary respectively among the odd primes and the positive integers.
For every k-component oriented link L, every partition 
of L into 
sublinks, and every labeling 
of such a partition, the number of 𝕏-colorings of any diagram of 
is a well-defined invariant of 
, of the form 
for some natural number 
. Letting 𝕏 and 
vary respectively in 
and among the labelings of 
, we define the derived invariant 
.
If 
is such that 
, we show that 
, where t(L) is the tunnel number of L, generalizing a result by Ishii. If 
is a "boundary partition" of L and 
denotes the infimum among the sums of the genera of a system of disjoint Seifert surfaces for the Lj's, then we show that 
. We point out further properties of 
, mostly in the case of 
, 
. By elaborating on a suitable version of a result by Inoue, we show that when L = K is a knot then 
, where 
is the breadth of the Alexander polynomial of K. However, for every g ≥ 1 we exhibit examples of genus-g knots having the same Alexander polynomial but different quandle invariants 
. Moreover, in such examples 
provides sharp lower bounds for the genera of the knots. On the other hand, we show that 
can give better lower bounds on the genus than 
, when L has k ≥ 2 components.
We show that in order to compute 
it is enough to consider only colorings with respect to the constant labeling 
. In the case when L = K is a knot, if either 
or 
provides a sharp lower bound for the knot genus, or if 
, then 
can be realized by means of the proper subfamily of quandles {𝕏 = (𝔽p, *)}, where p varies among the odd primes.
