Rational structure on algebraic tangles and closed incompressible surfaces in the complements of algebraically alternating knots and links
| dc.creator | Ozawa, Makoto | |
| dc.date | 2008-03-09 | |
| dc.date | 2009-05-07 | |
| dc.date.accessioned | 2026-07-07T13:11:59Z | |
| dc.date.available | 2026-07-07T13:11:59Z | |
| dc.description | Let $F$ be an incompressible, meridionally incompressible and not boundary-parallel surface with boundary in the complement of an algebraic tangle $(B,T)$. Then $F$ separates the strings of $T$ in $B$ and the boundary slope of $F$ is uniquely determined by $(B,T)$ and hence we can define the slope of the algebraic tangle. In addition to the Conway's tangle sum, we define a natural product of two tangles. The slopes and binary operation on algebraic tangles lead an algebraic structure which is isomorphic to the rational numbers. We introduce a new knot and link class, algebraically alternating knots and links, roughly speaking which are constructed from alternating knots and links by replacing some crossings with algebraic tangles. We give a necessary and sufficient condition for a closed surface to be incompressible and meridionally incompressible in the complement of an algebraically alternating knot or link $K$, in particular we show that if $K$ is a knot, then the complement of $K$ does not contain such a surface. | |
| dc.description | 15 pages, 15figures | |
| dc.identifier | https://arxiv.org/abs/0803.1302 | |
| dc.identifier | http://arxiv.org/abs/0803.1302 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/229455 | |
| dc.subject | Geometric Topology | |
| dc.subject | 57M25 (Primary) 57Q35 (Secondary) | |
| dc.title | Rational structure on algebraic tangles and closed incompressible surfaces in the complements of algebraically alternating knots and links | |
| dc.type | text |