论文标题
负极结和半通道多项式
Negative amphichiral knots and the half-Conway polynomial
论文作者
论文摘要
1979年，哈特利（Hartley）和川奇（Kawauchi）证明，强烈的两秒圆膜结构因子的conway多项式为$ f（z）f（-z）$。在本文中，我们将因子$ f（z）$归一化，以定义半程多项式。首先，我们证明半孔多项式满足了均等的绞线关系，从而提供了第一种可行的计算方法，我们用来计算具有12个或更少交叉点的结的半通道多项式。这种绞线的关系还导致对度数系数的图解解释，从中，我们从该系数上获得了对等效的伸出数字的下限。其次，我们完全表征了$ s^3 $的半通道多项式产生的多项式，回答了Hartley-Kawauchi的问题。作为一种特殊情况，我们构建了非切片的第一个例子，即用决定性的含量强烈的两性圆形结，回答了Manolescu的问题。这些结的双支封面在同源性恢复组中提供了潜在的非平凡扭转元件。
In 1979, Hartley and Kawauchi proved that the Conway polynomial of a strongly negative amphichiral knot factors as $f(z)f(-z)$. In this paper, we normalize the factor $f(z)$ to define the half-Conway polynomial. First, we prove that the half-Conway polynomial satisfies an equivariant skein relation, giving the first feasible computational method, which we use to compute the half-Conway polynomial for knots with 12 or fewer crossings. This skein relation also leads to a diagrammatic interpretation of the degree-one coefficient, from which we obtain a lower bound on the equivariant unknotting number. Second, we completely characterize polynomials arising as half-Conway polynomials of knots in $S^3$, answering a problem of Hartley-Kawauchi. As a special case, we construct the first examples of non-slice strongly negative amphichiral knots with determinant one, answering a question of Manolescu. The double branched covers of these knots provide potentially non-trivial torsion elements in the homology cobordism group.
