188 research outputs found
Morse-Novikov theory, Heegaard splittings and closed orbits of gradient flows
The works of Donaldson and Mark make the structure of the Seiberg-Witten
invariant of 3-manifolds clear. It corresponds to certain torsion type
invariants counting flow lines and closed orbits of a gradient flow of a
circle-valued Morse map on a 3-manifold. We study these invariants using the
Morse-Novikov theory and Heegaard splitting for sutured manifolds, and make
detailed computations for knot complements.Comment: 27 pages, 12 figure
Dehn surgeries on knots which yield lens spaces and genera of knots
Let be a hyperbolic knot in the 3-sphere. If -surgery on yields a
lens space, then we show that the order of the fundamental group of the lens
space is at most , where is the genus of . If we specialize to
genus one case, it will be proved that no lens space can be obtained from genus
one, hyperbolic knots by Dehn surgery. Therefore, together with known facts, we
have that a genus one knot admits Dehn surgery yielding a lens space if and
only if is the trefoil.Comment: 20 pages, 6 figure
Almost alternating diagrams and fibered links in S^3
Let be an oriented link with an alternating diagram . It is known that
is a fibered link if and only if the surface obtained by applying
Seifert's algorithm to is a Hopf plumbing. Here, we call a Hopf
plumbing if is obtained by successively plumbing finite number of Hopf
bands to a disk.
In this paper, we discuss its extension so that we show the following
theorem. Let be a Seifert surface obtained by applying Seifert's algorithm
to an almost alternating diagrams. Then is a fiber surface if and only if
is a Hopf plumbing.
We also show that the above theorem can not be extended to 2-almost
alternating diagrams, that is, we give examples of 2-almost alternating
diagrams for knots whose Seifert surface obtained by Seifert's algorithm are
fiber surfaces that are not Hopf plumbing. This is shown by using a criterion
of Melvin-Morton.Comment: 18 pages, 30 figure
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