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首页> 外文期刊>Journal of knot theory and its ramifications >Alexander quandle lower bounds for link genera
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Alexander quandle lower bounds for link genera

机译：链接生成器的Alexander Quandle下界

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Every finite field _q, q = p ~n, carries several Alexander quandle structures = (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. For every k-component oriented link L, every partition P of L into h: = P sublinks, and every labeling z? ε N ~h of such a partition, the number of X-colorings of any diagram of (L, z?) is a well-defined invariant of (L, P), of the form q ~(ax)(L, P, z)+1 for some natural number a _X(L, P, z?). Letting X and z 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, zmacr;). If P _M is such that P M = k, we show that A _Q(L, P _M)≤ t(L), where t(L) is the tunnel number of L, generalizing a result by Ishii. 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)≤ 2g(L, P)+2k-P-1. We point out further properties of A _Q(L, P), mostly in the case of A _Q(L): = A _Q(L, P _m), 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)≤ A(K), where A(K) 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 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 < 2 components. We show that in order to compute A _Q(L) it is enough to consider only colorings with respect to the constant labeling 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 primes.
机译：每个有限域_q，q = p〜n，带有几个Alexander quandle结构=（q，*）。我们用Q_F表示这些量子的族，其中p和n在奇质数和正整数之间分别变化。对于每个面向k分量的链接L，L的每个分区P都分为h：= P个子链接，每个标签z？ εN〜h这样的分区，任何图（L，z？）的X-色数都是（L，P）的明确定义的不变量，形式为q〜（ax）（L，P ，z）+1表示某个自然数a _X（L，P，z？）。令X和z分别在Q _F和P的标签之间变化，我们定义派生的不变式A _Q（L，P）：= sup {a _X（L，P，zmacr;）。如果P _M使得P M = k，我们证明A _Q（L，P _M）≤t（L），其中t（L）是L的隧道数，将Ishii的结果概括化。如果P是L的“边界分区”，并且g（L，P）表示L _j的不相交的Seifert曲面系统的总和中的最小值，那么我们证明A _Q（L，P）≤ 2g（L，P）+ 2k-P-1。我们指出A _Q（L，P）的其他属性，主要是在A _Q（L）的情况下：= A _Q（L，P _m），P _m = 1。井上，我们证明当L = K是一个结时，则A _Q（K）≤A（K），其中A（K）是K的亚历山大多项式的宽度。但是，对于每一个<1属g结，具有相同的亚历山大多项式，但具有不同的量子不变量A _Q。而且，在这样的示例中，A_Q为结的种类提供了尖锐的下界。另一方面，我们证明当L的k <2个分量时，A _Q（L）可以比A（L）给出更好的下界。我们表明，为了计算A _Q（L），仅考虑相对于常量标签z？的着色就足够了。 =1。在L = K为结的情况下，如果A _Q（K）= A（K）或A _Q（K）为结类提供了尖锐的下限，或者A _Q（K）=如图1所示，则可以通过适当的分位数{X =（F _p，*）}来实现A _Q（K），其中p在奇数素数之间变化。

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