科研之友
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摘要
Every finite field F-q, q = p(n), carries several Alexander quandle structures X = (F-q, *). We denote by Q(F) the family of these quandles, where p and n vary respectively among the odd primes and the positive integers. %26lt;br%26gt;For every k-component oriented link L, every partition P of L into h := vertical bar P vertical bar sublinks, and every labeling (z) over bar is an element of N-h of such a partition, the number of X-colorings of any diagram of (L, (z) over bar) is a well-defined invariant of (L, P), of the form q(aX)(L, P, ((z) over bar))(+ 1) for some natural number a(X)(L, P, (z) over bar ). Letting X and (z) over bar vary respectively in Q(F) and among the labelings of P, we define the derived invariant A(Q)(L, P) := sup{a(X)(L, P, (z) over bar)}. %26lt;br%26gt;If P-M is such that vertical bar P-M vertical bar = k, we show that A(Q)(L, P-M) %26lt;= t(L), where t(L) is the tunnel number of L, generalizing a result by Ishii. If P is a %26quot;boundary partition%26quot; 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%26apos;s, then we show that A(Q)(L, P) %26lt;= 2g(L, P)+ 2k -vertical bar P vertical bar- 1. We point out further properties of A(Q)(L, P), mostly in the case of A(Q)(L) := A(Q)(L, Pm), vertical bar P-m vertical bar = 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) %26lt;= A(K), where A(K) is the breadth of the Alexander polynomial of K. However, for every g %26gt;= 1 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, we show that A(Q)(L) can give better lower bounds on the genus than A(L), when L has k %26gt;= 2 components. %26lt;br%26gt;We show that in order to compute A(Q)(L) it is enough to consider only colorings with respect to the constant labeling (z) over bar = 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 primes.
- 出版日期2012-7
