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![A schematic depiction of τexch\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\tau _{\mathrm{exch}}$\end{document}. Here, we have deliberately chosen an asymmetric complex for CFK(K)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathit{CFK}(K)$\end{document} to better illustrate the switch map from CFK(K)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathit{CFK}(K)$\end{document} to CFK(Kr)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathit{CFK}(K^{r})$\end{document}](publication/383699675/figure/fig2/AS:11431281283957875@1729043391062/A-schematic-depiction-of-texchdocumentclass12ptminimal-usepackageamsmath.png)
A schematic depiction of τexch\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\tau _{\mathrm{exch}}$\end{document}. Here, we have deliberately chosen an asymmetric complex for CFK(K)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathit{CFK}(K)$\end{document} to better illustrate the switch map from CFK(K)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathit{CFK}(K)$\end{document} to CFK(Kr)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathit{CFK}(K^{r})$\end{document}
Source publication
![The swapping involution on K#Kr\documentclass[12pt]{minimal}...](publication/383699675/figure/fig1/AS:11431281283908073@1729043390903/The-swapping-involution-on-KKrdocumentclass12ptminimal-usepackageamsmath_Q320.jpg)
![A schematic depiction of τexch\documentclass[12pt]{minimal}...](publication/383699675/figure/fig2/AS:11431281283957875@1729043391062/A-schematic-depiction-of-texchdocumentclass12ptminimal-usepackageamsmath_Q320.jpg)
![The knot Floer complex CFK(K)\documentclass[12pt]{minimal}...](publication/383699675/figure/fig3/AS:11431281283975703@1729043391222/The-knot-Floer-complex-CFKKdocumentclass12ptminimal-usepackageamsmath_Q320.jpg)
![The knot Floer complex of K#Kr\documentclass[12pt]{minimal}...](publication/383699675/figure/fig4/AS:11431281283908074@1729043391366/The-knot-Floer-complex-of-KKrdocumentclass12ptminimal-usepackageamsmath_Q320.jpg)
![The cobordism W1,n\documentclass[12pt]{minimal} \usepackage{amsmath}...](publication/383699675/figure/fig5/AS:11431281283918153@1729043391530/The-cobordism-W1-ndocumentclass12ptminimal-usepackageamsmath-usepackagewasysym_Q320.jpg)
We prove that the (2,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(2,1)$\end{document}-cable of the figure-eight knot is not smoothly slice by showing that its bran...
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Citations
We prove a formula for the involutive concordance invariants of the cabled knots in terms of those of the companion knot and the pattern knot. As a consequence, we show that any iterated cable of a knot with parameters of the form (odd,1) is not smoothly slice as long as either of the involutive concordance invariants of the knot is nonzero. Our formula also gives new bounds for the unknotting number of a cabled knot, which are sometimes stronger than other known bounds coming from knot Floer homology.
We introduce L-functions attached to negative-definite plumbed manifolds as the Mellin transforms of homological blocks. We prove that they are entire functions and their values at s=0 are equal to the Witten–Reshetikhin–Turaev invariants by using asymptotic techniques developed by the author in the previous papers. We also prove linear relations between special values at negative integers of some L-functions, which are common generalizations of Hurwitz zeta functions, Barnes zeta functions and Epstein zeta functions.
These notes follow a lecture series at the "Singularities and low dimensional topology" winter school at the R\'enyi Institute in January 2023, with a target audience of graduate students in singularity theory and low-dimensional topology. The lectures discuss the basics of four-dimensional manifold topology, connecting this rich subject to knot theory on one side and to contact, symplectic, and complex geometry (through Stein surfaces) on the other side of the spectrum.
Although exotic blow-ups of the projective plane at n points have been constructed for every , the only examples known by means of rational blowdowns satisfy . It has been an intriguing problem whether it is possible to decrease n. In this paper, we construct the first exotic with this technique. We also construct exotic for . All of them are minimal and symplectic, as they are produced from projective surfaces W with Wahl singularities and big and nef. In more generality, we elaborate on the problem of finding exotic from these Kollár–Shepherd-Barron–Alexeev surfaces W, obtaining explicit geometric obstructions on the corresponding configurations of rational curves.
