We define two concordance invariants of knots using framed instanton homology. These invariants ${\nu }^{♯}$ and ${\tau }^{♯}$ provide bounds on slice genus and maximum self-linking number, and the latter is a concordance homomorphism which agrees in all known cases with the $\tau$ invariant in Heegaard Floer homology. We use ${\nu }^{♯}$ and ${\tau }^{♯}$ to compute the framed instanton homology of all nonzero rational Dehn surgeries on: 20 of the 35 nontrivial prime knots through eight crossings, infinite families of twist and pretzel knots, and instanton L-space knots; and of 19 of the first 20 closed hyperbolic manifolds in the Hodgson–Weeks census. In another application, we determine when the cable of a knot is an instanton L-space knot. Finally, we discuss applications to the spectral sequence from odd Khovanov homology to the framed instanton homology of branched double covers, and to the behaviors of ${\tau }^{♯}$ and $\tau$ under genus-2 mutation.

中文翻译：

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框架瞬时同源性和一致性

我们使用框架瞬时同调定义了两个节点的一致性不变量。这些不变量${\nu }^{♯}$和${\tau }^{♯}$提供切片属和最大自链接数的界限，后者是一致同态，在所有已知情况下都与$\tau$Heegaard Floer 同调不变。我们用${\nu }^{♯}$和${\tau }^{♯}$计算所有非零理性 Dehn 手术的框架瞬时同源性： 35 个非平凡素数结中的 20 个通过 8 个交叉点、无限系列的扭曲和椒盐卷饼结，以及瞬时 L 空间结；Hodgson-Weeks 人口普查中前 20 个闭合双曲流形中的 19 个。在另一个应用程序中，我们确定一个结的电缆何时是一个瞬时 L 空间结。最后，我们讨论了从奇 Khovanov 同源到分支双覆盖的框架瞬时同源的光谱序列的应用，以及${\tau }^{♯}$和$\tau$在 genus-2 突变下。