We show that the link cobordism maps defined by the author are graded and satisfy a grading change formula. Using the grading change formula, we prove a new bound for $\Upsilon_K(t)$ for knot cobordisms in negative definite 4-manifolds. As another application, we show that the link cobordism maps associated to a connected, closed surface in $S^4$ are determined by the genus of the surface. We also prove a new adjunction relation and adjunction inequality for the link cobordism maps. Along the way, we see how many known results in Heegaard Floer homology can be proven using basic properties of the link cobordism maps, together with the grading change formula.

中文翻译：

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链接 Floer 同源性中的链接 cobordisms 和绝对分级

我们表明作者定义的链接cobordism 地图被分级并满足分级变化公式。使用分级变化公式，我们证明了 $\Upsilon_K(t)$ 对于负定 4-流形中的结点 cobordisms 的新界限。作为另一个应用，我们展示了与 $S^4$ 中连接的封闭曲面相关联的链接共边映射是由曲面的属确定的。我们还证明了链接协同映射的新附属关系和附属不等式。在此过程中，我们看到了 Heegaard Floer 同源性中的多少已知结果可以使用链接协同图的基本属性以及分级变化公式来证明。