268 research outputs found
Knot adjacency and satellites
A knot K is called n-adjacent to the unknot, if K admits a projection
containing n generalized crossings such that changing any m (no larger than n)
of them yields a projection of the unknot. We show that a non-trivial satellite
knot K is n-adjacent to the unknot, for some n>0, if and only if it is
n-adjacent to the unknot in any companion solid torus. In particular, every
model knot of K is n-adjacent to the unknot. Along the way of proving these
results, we also show that 2-bridge knots of the form K_{p/q}, where
p/q=[2q_1,2q_2] for some integers q_1,q_2, are precisely those knots that have
genus one and are 2-adjacent to the unknot.Comment: 13 pages, 3 figures. to appear in Topology and Its Application
Crosscap numbers and the Jones polynomial
We give sharp two-sided linear bounds of the crosscap number (non-orientable
genus) of alternating links in terms of their Jones polynomial. Our estimates
are often exact and we use them to calculate the crosscap numbers for several
infinite families of alternating links and for several alternating knots with
up to twelve crossings. We also discuss generalizations of our results for
classes of non-alternating links.Comment: 27 pages. Minor corrections and modifications. To appear in Advances
of Mathematic
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