Symplectic structure on colorings, Lagrangian tangles and Tits buildings

Jan Dymara ¹   ; Tadeusz Januszkiewicz ²   ; Józef H. Przytycki ³

¹ Instytut Matematyczny Uniwersytet Wrocławski Wrocław, Poland <a href="https://orcid.org/0000-0002-7127-3298">ORCID: 0000-0002-7127-3298</a>
² Instytut Matematyczny PAN Warszawa, Poland and Instytut Matematyczny Uniwersytet Wrocławski Wrocław, Poland <a href="https://orcid.org/0000-0002-6570-0281">ORCID: 0000-0002-6570-0281</a>
³ Department of Mathematics The George Washington University Washington, DC, U.S.A. and Institute of Mathematics University of Gdańsk Gdańsk, Poland <a href="https://orcid.org/0000-0002-1600-8889">ORCID: 0000-0002-1600-8889</a>

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 68 (2020) no. 2, pp. 169-194

Résumé

We define a symplectic form $\widehat \varphi $ on a free $R$-module $ R^{2n-2}$ associated to $2n$ points on a circle. Using this form, we establish a relation between submodules of $R^{2n-2}$ induced by Fox $R$-colorings of an $n$-tangle and Lagrangians or virtual Lagrangians in the symplectic structure $( R^{2n-2},\widehat \varphi ) $ depending on whether $R$ is a field or a PID. We prove that when $R=\mathbb {Z}_{p}$, $p \gt 2$, all Lagrangians are induced by Fox $R$-colorings of some $n$-tangles and note that for $p=2$ and $n \gt 3$ this is no longer true. For any ring, every $2\pi /n$-rotation of an $n$-tangle yields an isometry of the symplectic space $R^{2n-2}$. We analyze invariant Lagrangian subspaces of this rotation and we partially answer the question whether an operation of rotation (generalized mutation) defined by Anstee et al. (1989) preserves the first homology group of the double branched cover of $S^{3}$ along a given link.

Détail

DOI : 10.4064/ba201230-2-3

Keywords: define symplectic form widehat varphi r module n associated points circle using form establish relation between submodules n induced fox r colorings n tangle lagrangians virtual lagrangians symplectic structure n widehat varphi depending whether field pid prove mathbb lagrangians induced fox r colorings n tangles note longer ring every n rotation n tangle yields isometry symplectic space nbsp n analyze invariant lagrangian subspaces rotation partially answer question whether operation rotation generalized mutation defined cite a p r preserves first homology group double branched cover along given link