scholarly journals From Goeritz Matrices to Quasi-alternating Links

2011 ◽  
pp. 257-316 ◽  
Author(s):  
Józef H. Przytycki
Keyword(s):  
Alternating Links

scholarly journals Almost alternating links

1992 ◽  
Vol 46 (2) ◽  
pp. 151-165 ◽  
Author(s):  
Colin C. Adams ◽  
Jeffrey F. Brock ◽  
John Bugbee ◽  
Timothy D. Comar ◽  
Keith A. Faigin ◽  
...  
Keyword(s):  
Alternating Links

scholarly journals Braid index bounds ropelength from below

2020 ◽  
Vol 29 (04) ◽  
pp. 2050019
Author(s):  
Yuanan Diao
Keyword(s):  
Lower Bound ◽  
General Formula ◽  
Crossing Number ◽  
Crossing Numbers ◽  
Braid Index ◽  
Alternating Links

For an unoriented link [Formula: see text], let [Formula: see text] be the ropelength of [Formula: see text]. It is known that in general [Formula: see text] is at least of the order [Formula: see text], and at most of the order [Formula: see text] where [Formula: see text] is the minimum crossing number of [Formula: see text]. Furthermore, it is known that there exist families of (infinitely many) links with the property [Formula: see text]. A long standing open conjecture states that if [Formula: see text] is alternating, then [Formula: see text] is at least of the order [Formula: see text]. In this paper, we show that the braid index of a link also gives a lower bound of its ropelength. More specifically, we show that there exists a constant [Formula: see text] such that [Formula: see text] for any [Formula: see text], where [Formula: see text] is the largest braid index among all braid indices corresponding to all possible orientation assignments of the components of [Formula: see text] (called the maximum braid index of [Formula: see text]). Consequently, [Formula: see text] for any link [Formula: see text] whose maximum braid index is proportional to its crossing number. In the case of alternating links, the maximum braid indices for many of them are proportional to their crossing numbers hence the above conjecture holds for these alternating links.

On a Structural Property of the Groups of Alternating Links

1976 ◽  
Vol 28 (3) ◽  
pp. 568-588 ◽  
Author(s):  
E. J. Mayland ◽  
Kunio Murasugi
Keyword(s):  
Structural Property ◽  
Prime Power ◽  
Residually Finite ◽  
Alternating Links ◽  
Knot Polynomial

In this paper, we will prove, as a consequence of the main theorem,THEOREM A. (See Corollary 2.6). The group of an alternating knot, for which the leading coefficient of the knot polynomial is a prime power, is residually finite and solvable.

scholarly journals The Kauffman polynomial of alternating links

1995 ◽  
Vol 65 (3) ◽  
pp. 229-236 ◽  
Author(s):  
Yoshiyuki Yokota
Keyword(s):  
Kauffman Polynomial ◽  
Alternating Links

scholarly journals The colored Kauffman Skein relation and the head and tail of the colored Jones polynomial

2017 ◽  
Vol 26 (03) ◽  
pp. 1741002 ◽  
Author(s):  
Mustafa Hajij
Keyword(s):  
Natural Extension ◽  
Jones Polynomial ◽  
Colored Jones Polynomial ◽  
Alternating Links ◽  
Skein Relation

Using the colored Kauffman skein relation, we study the highest and lowest [Formula: see text] coefficients of the [Formula: see text] unreduced colored Jones polynomial of alternating links. This gives a natural extension of a result by Kauffman in regard with the Jones polynomial of alternating links and its highest and lowest coefficients. We also use our techniques to give a new and natural proof for the existence of the tail of the colored Jones polynomial for alternating links.

POSITIVE ALTERNATING LINKS ARE POSITIVELY ALTERNATING

2000 ◽  
Vol 09 (01) ◽  
pp. 107-112 ◽  
Author(s):  
TAKUJI NAKAMURA
Keyword(s):  
Alternating Links ◽  
Alternating Link

By using a result of P. R. Cromwell on homogeneous links which contain both positive links and alternating links, we prove that any reduced alternating diagram of a positive alternating link is positive.

scholarly journals KAUFFMAN–HARARY CONJECTURE HOLDS FOR MONTESINOS KNOTS

2004 ◽  
Vol 13 (04) ◽  
pp. 467-477 ◽  
Author(s):  
MARTA M. ASAEDA ◽  
JÓZEF H. PRZYTYCKI ◽  
ADAM S. SIKORA
Keyword(s):  
Double Cover ◽  
Incompressible Surfaces ◽  
Pretzel Knots ◽  
Prime Determinant ◽  
Alternating Links

The Kauffman–Harary conjecture states that for any reduced alternating diagram K of a knot with a prime determinant p, every non-trivial Fox p-coloring of K assigns different colors to its arcs. We generalize this conjecture by stating it in terms of homology of the double cover of S3. In this way we extend the scope of the conjecture to all prime alternating links of arbitrary determinants. We first prove the Kauffman–Harary conjecture for pretzel knots and then we generalize our argument to show the generalized Kauffman–Harary conjecture for all Montesinos links. Finally, we discuss on the relation between the conjecture and Menasco's work on incompressible surfaces in exteriors of alternating links.

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