2 research outputs found
Branched covers of contact manifolds
We will discuss what is known about the construction of contact structures via branched covers, emphasizing the search for universal transverse knots. Recall that a topological knot is called universal if all 3-manifold can be obtained as a cover of the 3-sphere branched over that knot. Analogously one can ask if there is a transverse knot in the standard contact structure on S³ from which all contact 3-manifold can be obtained as a branched cover over this transverse knot. It is not known if such a transverse knot exists.Ph.D
Intrinsic symmetry groups of links with 8 and fewer crossings
We present an elementary derivation of the "intrinsic" symmetry groups for
knots and links of 8 or fewer crossings. The standard symmetry group for a link
is the mapping class group \MCG(S^3,L) or \Sym(L) of the pair .
Elements in this symmetry group can (and often do) fix the link and act
nontrivially only on its complement. We ignore such elements and focus on the
"intrinsic" symmetry group of a link, defined to be the image of
the natural homomorphism \MCG(S^3,L) \rightarrow \MCG(S^3) \cross \MCG(L).
This different symmetry group, first defined by Whitten in 1969, records
directly whether is isotopic to a link obtained from by permuting
components or reversing orientations.
For hyperbolic links both \Sym(L) and can be obtained using the
output of \texttt{SnapPea}, but this proof does not give any hints about how to
actually construct isotopies realizing . We show that standard
invariants are enough to rule out all the isotopies outside for all
links except , and where an additional construction
is needed to use the Jones polynomial to rule out "component exchange"
symmetries. On the other hand, we present explicit isotopies starting with the
positions in Cerf's table of oriented links which generate for each
link in our table. Our approach gives a constructive proof of the
groups.Comment: 72 pages, 66 figures. This version expands the original introduction
into three sections; other minor changes made for improved readabilit