论文标题
Chern-simons功能，奇异的激体和四维扣子
Chern-Simons functional, singular instantons, and the four-dimensional clasp number
论文作者
论文摘要
Kronheimer和Mrowka询问四维扣子数和切片属之间的差异是否可以任意大。通过研究源自iporiast单数插入理论的结结，并且与Chern-Simons功能密切相关，从而肯定了这个问题。这也回答了利文斯顿关于切片数字的猜想。还研究了一个单一的InstantonFrøyshov结。如果用整数系数定义，则为无定向的切片属提供了下限，并计算用于准偏置和圆环结。相比之下，对于某些其他系数环，不变式的结构均具有结的倍数。该结果用于解决Poudel和Saveliev的猜想，内容涉及圆环结的无可用$ su（2）$。此外，对于与非零签名的结之间的一致性，可以表明，一致性补体的无可迹象表达限制了结基团的非平凡表示。最后，提供了一些向米裂猜想扩展到圆环结的证据。
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.
