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![If a knot K′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K'$$\end{document} can be obtained from K by mH(n)-moves, then K′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K'$$\end{document} can also be obtained from K by m(n-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m(n-1)$$\end{document}H(2)-moves, each of which is realized by the attaching of a noncoherent band. If the roots of the bands are gathered as shown, the totality of all m(n-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m(n-1)$$\end{document} band moves is accomplished by a single H(m(n-1)+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H(m(n-1)+1)$$\end{document}-move, the one inside the dashed oval. Observe that the number of arcs (colored in red in a) inside the dashed oval equals 1 plus the number of bands](publication/358243228/figure/fig3/AS:1142765186494464@1649467860155/If-a-knot-Kdocumentclass12ptminimal-usepackageamsmath-usepackagewasysym.png)
If a knot K′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K'$$\end{document} can be obtained from K by mH(n)-moves, then K′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K'$$\end{document} can also be obtained from K by m(n-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m(n-1)$$\end{document}H(2)-moves, each of which is realized by the attaching of a noncoherent band. If the roots of the bands are gathered as shown, the totality of all m(n-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m(n-1)$$\end{document} band moves is accomplished by a single H(m(n-1)+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H(m(n-1)+1)$$\end{document}-move, the one inside the dashed oval. Observe that the number of arcs (colored in red in a) inside the dashed oval equals 1 plus the number of bands
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![An H(n)-move can be realized by n-1\documentclass[12pt]{minimal}...](publication/358243228/figure/fig2/AS:1142765186486272@1649467860136/An-Hn-move-can-be-realized-by-n-1documentclass12ptminimal-usepackageamsmath_Q320.jpg)
![If a knot K′\documentclass[12pt]{minimal} \usepackage{amsmath}...](publication/358243228/figure/fig3/AS:1142765186494464@1649467860155/If-a-knot-Kdocumentclass12ptminimal-usepackageamsmath-usepackagewasysym_Q320.jpg)
We define a broad class of graphs that generalize the Gordian graph of knots. These knot graphs take into account unknotting operations, the concordance relation, and equivalence relations generated by knot invariants. We prove that overwhelmingly, the knot graphs are not Gromov hyperbolic, with the exception of a particular family of quotient knot...
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![D~n\documentclass[12pt]{minimal} \usepackage{amsmath}...](publication/377630695/figure/fig1/AS:11431281226326278@1709174929246/Dndocumentclass12ptminimal-usepackageamsmath-usepackagewasysym_Q320.jpg)
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Citations
We study the C P 2 \mathbb {CP}^2 -slicing number of knots, i.e. the smallest m ≥ 0 m\geq 0 such that a knot K ⊆ S 3 K\subseteq S^3 bounds a properly embedded, null-homologous disk in a punctured connected sum ( # m C P 2 ) × (\#^m\mathbb {CP}^2)^{\times } . We find knots for which the smooth and topological C P 2 \mathbb {CP}^2 -slicing numbers are both finite, nonzero, and distinct. To do this, we give a lower bound on the smooth C P 2 \mathbb {CP}^2 -slicing number of a knot in terms of its double branched cover and an upper bound on the topological C P 2 \mathbb {CP}^2 -slicing number in terms of the Seifert form.
We study the -slicing number of knots, i.e. the smallest such that a knot bounds a properly embedded, null-homologous disk in a punctured connected sum . We give a lower bound on the smooth -slicing number of a knot in terms of its double branched cover, and we find knots with arbitrarily large but finite smooth -slicing number. We also give an upper bound on the topological -slicing number in terms of the Seifert form and find knots for which the smooth and topological -slicing numbers are both finite, nonzero, and distinct.
