Chern-Simons functional, singular instantons, and the four-dimensional clasp number
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| Format: | Preprint |
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2020
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| _version_ | 1866916383747473408 |
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| author | Daemi, Aliakbar Scaduto, Christopher |
| author_facet | Daemi, Aliakbar Scaduto, Christopher |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2007_13160 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Chern-Simons functional, singular instantons, and the four-dimensional clasp number Daemi, Aliakbar Scaduto, Christopher Geometric Topology 57R58 57M05 57K18 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. |
| title | Chern-Simons functional, singular instantons, and the four-dimensional clasp number |
| topic | Geometric Topology 57R58 57M05 57K18 |
| url | https://arxiv.org/abs/2007.13160 |