Kronheimer and Mrowka introduced a new knot invariant, called $s^{♯}$, which is a gauge theoretic analogue of Rasmussen's $s$ invariant. In this article, we compute Kronheimer and Mrowka's invariant for some classes of knots, including algebraic knots and the connected sums of quasi‐positive knots with non‐trivial right‐handed torus knots. These computations reveal some unexpected phenomena: we show that $s^{♯}$ does not have to agree with $s$, and that $s^{♯}$ is not additive under connected sums of knots.

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关于Kronheimer-Mrowka协和不变量

Kronheimer和Mrowka引入了一个新的结不变式，称为 $s^{♯}$，这是拉斯穆森（Rasmussen）的规范理论类似物 $s$不变的 在本文中，我们计算某些类结的Kronheimer和Mrowka不变式，包括代数结和准正结与非平凡右手环结的连接和。这些计算揭示了一些意想不到的现象：我们证明了$s^{♯}$ 不必同意 $s$， 然后 $s^{♯}$ 在连接的总和下不是可加的。