Alexander quandle lower bounds for link genera

TL;DR

This paper introduces a new quandle-based invariant for links that provides lower bounds on tunnel and genus numbers, improving understanding of link complexity and knot genus, with computational simplifications and comparisons to classical invariants.

Contribution

It defines a novel Alexander quandle invariant A_Q for links, establishing bounds on tunnel number and genus, and demonstrates its effectiveness and computational aspects compared to traditional invariants.

Findings

A_Q(L) bounds the tunnel number from below.

A_Q(L) provides sharp lower bounds for knot genus.

A_Q(L) can outperform classical Alexander polynomial bounds.

Abstract

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.