ALEXANDER QUANDLE LOWER BOUNDS FOR LINK GENERA

Abstract

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