501 research outputs found
Relationships between braid length and the number of braid strands
For a knot K, let b_n(K) be the minimum length of an n-stranded braid
representative of K. Examples of knots exist for which b_n(K) is a
non-increasing function. We investigate the behavior of b_n(K). We develop
bounds on the function in terms of the genus of K, with stronger results for
homogeneous knots and braid positive knots. For knots of nine or fewer
crossings, we show that b_n(K) is an increasing function and determine it
completely.Comment: 9 pages, 2 figures; minor revision
Euclidean Mahler measure and twisted links
If the twist numbers of a collection of oriented alternating link diagrams
are bounded, then the Alexander polynomials of the corresponding links have
bounded euclidean Mahler measure (see Definition 1.2). The converse assertion
does not hold. Similarly, if a collection of oriented link diagrams, not
necessarily alternating, have bounded twist numbers, then both the Jones
polynomials and a parametrization of the 2-variable Homflypt polynomials of the
corresponding links have bounded Mahler measure.Comment: This is the version published by Algebraic & Geometric Topology on 7
April 200
Homogeneous links, Seifert surfaces, digraphs and the reduced Alexander polynomial
We give a geometric proof of the following result of Juhasz. \emph{Let
be the leading coefficient of the Alexander polynomial of an alternating knot
. If then has a unique minimal genus Seifert surface.} In
doing so, we are able to generalise the result, replacing `minimal genus' with
`incompressible' and `alternating' with `homogeneous'. We also examine the
implications of our proof for alternating links in general.Comment: 37 pages, 28 figures; v2 Main results generalised from alternating
links to homogeneous links. Title change
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