陈-西蒙斯泛函、奇异瞬子与四维 clasp 数
Chern-Simons functional, singular instantons, and the four-dimensional clasp number
AI总结：
本文利用源自等变奇异瞬子理论并与陈-西蒙斯泛函相关的纽结不变量，肯定回答了四维 clasp 数与切片亏格之差可否任意大的问题，同时解决 Livingston 猜想，并研究奇异瞬子 Frøyshov 不变量及其在无向切片亏格、环面纽结表示和协边中的应用。
摘要：
Kronheimer 和 Mrowka 曾问：四维 clasp 数与切片亏格之间的差是否可以任意大。本文通过研究一个源自等变奇异瞬子理论、且与陈-西蒙斯泛函密切相关的纽结不变量，对这一问题给出了肯定回答。这一结果同时也回答了 Livingston 关于 slicing 数的一个猜想。本文还研究了纽结的奇异瞬子 Frøyshov 不变量。若用整数系数定义，该不变量给出无向切片亏格的一个下界，并且本文对 quasi-alternating 纽结和环面纽结计算了该不变量。与之相对，对于某些其他系数环，该不变量被识别为纽结签名的某个倍数。这一结果被用来处理 Poudel 和 Saveliev 关于环面纽结的无迹 $SU(2)$ 表示的一个猜想。此外，对于签名非零的纽结之间的协边，本文证明协边补空间上存在一个无迹表示，该表示限制到各纽结群上时给出非平凡表示。最后，本文提供了一些证据，支持将 slice-ribbon 猜想推广到环面纽结。
英文摘要：
Kronheimer and Mrowka asked whether the difference between the four-dimensional clasp number and the slice genus can be arbitrarily large. This question is answered affirmatively by studying a knot invariant derived from equivariant singular instanton theory, and which is closely related to the Chern--Simons functional. This also answers a conjecture of Livingston about slicing numbers. Also studied is the singular instanton Frøyshov invariant of a knot. If defined with integer coefficients, this gives a lower bound for the unoriented slice genus, and is computed for quasi-alternating and torus knots. In contrast, for certain other coefficient rings, the invariant is identified with a multiple of the knot signature. This result is used to address a conjecture by Poudel and Saveliev about traceless $SU(2)$ representations of torus knots. Further, for a concordance between knots with non-zero signature, it is shown that there is a traceless representation of the concordance complement which restricts to non-trivial representations of the knot groups. Finally, some evidence towards an extension of the slice-ribbon conjecture to torus knots is provided.
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