Title : Geometry of knots and links
Speaker : Alexander Mednykh (Novosibirsk State University)
Abstract :

We consider knots and links as a singular set of hyperbolic cone--manifolds with the three-sphere as underlying space. Then, in many important cases, the lengths of singular geodesics and cone angles are related with each other by number of relations similar to the classical Sine, Cosine and Tangent rules. In particular, the following results take a place \begin{theorem} Let $B(\alpha, \beta, \gamma),\,\,0<\alpha, \beta, \gamma<\pi $ be a hyperbolic Borromean rings cone--manifold. Denote by $l_\alpha, \,l_\beta,\,l_\gamma$ the lengths of singular geodesics of $B(\alpha, \beta, \gamma)$ with cone angles $\alpha, \,\beta,\, \gamma,$ respectively. Then we have $$ \frac{\tan{\frac{\alpha}2}} {\tanh{\frac{l_\alpha}4}} = \frac{\tan{\frac\beta 2}}{\tanh{\frac{l_\beta} 4 }}= \frac{\tan{\frac{\gamma}2}} {\tanh{\frac{l_\gamma}4}}\,\,\, ({\rm The\,\, Tangent\,\, Rule})$$ and $$ \frac{\sin{\frac{\alpha}2}} {\sinh{\frac{l_\alpha}4}} \,\frac{\sin{\frac\beta 2}}{\sinh{\frac{l_\beta} 4 }} \,\frac{\cos {\frac{\gamma}2}} {\cosh{\frac{l_\gamma}4}} = 1\,\,\, ({\rm The\,\,Sine-Cosine\,\, Rule}). $$ \end{theorem} \bigskip \begin{theorem} Let $W(\alpha, \beta)\,$ be a hyperbolic Whitehead cone--manifold. Denote by $\gamma_\alpha,\, \gamma_\beta$ the complex lengths of singular geodesics of $W(\alpha, \beta)$ with cone angles $\alpha,\, \beta,$ respectively. Then we obtain $$ \frac{\tan{\frac{\alpha}2}} {\tanh{\frac{\gamma_\alpha}4}} = \frac{\tan{\frac\beta 2}}{\tanh{\frac{\gamma_\beta} 4 }}\,\,\, ({\rm The\,\, Tangent\,\, Rule}).$$ \end{theorem} Similar results as well the Sine and Cosine rules are obtained in [3] for the the twist link cone--manifolds $\frac{4k+4}{2k+1}(\alpha,\,\beta),\,k=1,2,3,\ldots.$ Follow Rolfsen denote by $6^2_2$ the two bridge link with a slope $10/3.$ \bigskip \begin{theorem} Let $6^2_2(\alpha, \beta)\,$ be a hyperbolic cone--manifold. Denote by $l_\alpha,\, l_\beta$ the lengths of singular geodesics of $6^2_2(\alpha,\, \beta)$ with cone angles $\alpha, \,\beta,$ respectively. Then we have $$ \frac{\sin{\frac{\alpha}2}} {\sinh{\frac{l_\alpha}2}} =\frac{\sin{\frac\beta 2}}{\sinh{\frac{l_\beta} 2 }}\,\,\, ({\rm The\,\,Sine\,\, Rule}) $$ and $$ \frac{\cos\frac{\alpha}2\cosh\frac{l_\beta}2- \cos\frac{\beta}2\cosh\frac{l_\alpha}2 }{\cos\alpha-\cos\beta}= 1\,\, ({\rm The\,\,Cosine\,\, Rule}). $$ \end{theorem} Similar results are expected to be true for any two bridge cone-- manifold $$(2k+1+\frac{1}{2k+1})(\alpha,\, \beta),\,k=1,2,3,\ldots.$$ \begin{thebibliography}{99} \bibitem[1]{1} A.~D.~Mednykh, {\em On the remarkable properties of the hyperbolic Whitehead link cone--manifold,} Knots in Hellas'98 (C.~McA.~Gordon, V.~F.~R.~Jones, L.~H.~Kauffman, S.~Lambropoulou, J.~H.~Przytycki, Eds.), Series Knots and Everything, Singapore et al.: World Scientific, 2000, Vol. 24, 290--305. \bibitem[2]{2} A.~D.~Mednykh, {\em On hyperbolic and spherical volumes for link cone--manifolds,} Kleinian Groups and Hyperbolic 3--manifolds, Proceedings of the Warwick Workshop, September 2001, Lond. Math. Soc. Lec. Notes, 299, (Y.~Komori, V.~Markovic and C.~, Eds.), Cambridge Univ. Press, 2003, 145--163. \bibitem[3]{3} D.~A.~Derevnin, A.~D.~Mednykh and M.~Mulazzani, {\em Volumes for twist link cone--manifolds,} Proceedings of the Conference in Low--dimensional Topology, Medina, 2002, Special volume in honor of Fico Gonzalez-Acuna, to appear. \end{thebibliography}

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