Alexander polynomials of alternating knots of genus two III

Jong, In Dae

doi:10.1016/j.topol.2011.11.014

Cited by 10 publications

(17 citation statements)

References 17 publications

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“…This conjecture is known to be true for many classes of links including the class of 2-bridge links, and more recently proved for all alternating knots of genus up to 2 by P. Ozsváth and Z. Szabó [18] and I. Jong [5]. On the other hand, K. Murasugi [17] and I. Jong [5] observed that there are alternating Laurent polynomials of degree 4 which are realized by knots and satisfy the trapezoidal conjecture, but are not realizable by any alternating knots.…”

Section: Algebraic Alternation Number and Algebraic τ -Alternation Numentioning

confidence: 87%

“…Since |σ (K ) − s(K )| is a concordance invariant, Abe's inequality actually implies the inequality alt[K ] σ (K ) − s(K ) /2, which is useful to know the value alt[K ], although Abe's inequality does not detect the assertions of Theorems 3.1 and 3.3 because of its concordance invariance. For example, let T m be the m-fold connected sum of T (4,5). Then we have alt(T m ) = alt[T m ] = 2m by Kanenobu's calculation.…”

Section: Remark 34mentioning

confidence: 98%

“…On the other hand, K. Murasugi [17] and I. Jong [5] observed that there are alternating Laurent polynomials of degree 4 which are realized by knots and satisfy the trapezoidal conjecture, but are not realizable by any alternating knots. For example, the Alexander polynomial A(9 44 ; t) = 1 − 4t + 7t 2 − 4t 3 + t 4 is such an example, in addition satisfying the Ozsváth-Szabó condition in [18].…”

Section: Algebraic Alternation Number and Algebraic τ -Alternation Numentioning

confidence: 99%

“…[19,20]). By using C. Livingston's observation in [15], T. Abe observed in [1] the inequality alt(K ) |σ (K ) − s(K )|/2, by which T. Abe proved that every torus knot K = T (p, q) (p > 2, q > 3) except (p, q) = (3, 4), (3,5) has alt[K ] > 1, meaning that the almost alternating torus knots are just T (3,4) and T (3,5), confirming a conjecture by C. Adams in [2]. For further calculations on the alternation numbers of torus knots, see T. Kanenobu [6].…”

Section: Remark 34mentioning

confidence: 99%

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On alternation numbers of links

Kawauchi

2010

Topology and its Applications

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We construct infinitely many hyperbolic links with x-distance far from the set of (possibly, splittable) alternating links in the concordance class of every link. A sensitive result is given for the concordance class of every (possibly, split) alternating link. Our proof uses an estimate of the τ -distance by an Alexander invariant and the topological imitation theory, both established earlier by the author. Alternation number and τ -alternation numberIn this paper, links are always oriented links in the oriented 3-sphere S 3 and a knot is regarded as a link of one component. An alternating link is a link with an alternating diagram, a link diagram such that an over-crossing and an under-crossing appear alternatively along every knot diagram component. Let A be the set of (possibly, splittable) alternating links. After the solution of the Tait flype conjecture on alternating links by W.W. Menasco and M.B. Thistlethwaite in [16], it became an important question to ask how a non-alternating link is "close to" or "far from" the set A under a suitable metric. The x-distance (or Gordian distance) d x (L, L ) between links L and L with the same number of components is the minimal number of cross-changes transforming a diagram of L into a diagram of L . A zero-linking twist of an oriented link L is an operation on links to obtain a link L from L by a twist along a trivial knot O such that L ∩ O = ∅ and the linking number Link(L, O ) = 0, and the τ -distance d τ (L, L ) between links L and L with the same number of components is the minimal number of zero-linking twists transforming L into L (cf. [12]). For links L, L with different numbers of components, we define d τ (L, L ) = d(L, L ) = +∞. Since the crossing change is a zero-linking twist and every link diagram is transformed into a diagram of a trivial link by crossing changes, we have 0 d τ (L, L ) d x (L, L ) < +∞ for all links L, L with the same number of components. The x-distance and the τ -distance are metric functions on the set of links with r components for every r 1. An estimate of d τ (L, L ) was done in [12] in terms of local link signatures and

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Section: Algebraic Alternation Number and Algebraic τ -Alternation Numentioning

confidence: 87%

Section: Remark 34mentioning

confidence: 98%

Section: Algebraic Alternation Number and Algebraic τ -Alternation Numentioning

confidence: 99%

Section: Remark 34mentioning

confidence: 99%

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On alternation numbers of links

Kawauchi

2010

Topology and its Applications

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“…The trapezoidal conjecture has been verified for several classes of alternating knots, e.g. for algebraic alternating knots by Murasugi [19] and alternating knots of genus two by Ozsváth and Szabó [23] and Jong [12].…”

Section: Conjecture 4 ([4]mentioning

confidence: 93%

On coxeter mapping classes and fibered alternating links

Hironaka¹,

Liechti²

2016

Michigan Math. J.

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Alternating-sign Hopf plumbing along a tree yields fibered alternating links whose homological monodromy is, up to a sign, conjugate to some alternating-sign Coxeter transformation. Exploiting this tie, we obtain results about the location of zeros of the Alexander polynomial of the fibered link complement implying a strong case of Hoste's conjecture, the trapezoidal conjecture, bi-orderability of the link group, and a sharp lower bound for the homological dilatation of the monodromy of the fibration. The results extend to more general hyperbolic fibered 3-manifolds associated to alternating-sign Coxeter graphs.

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Alexander and Jones polynomials of weaving 3-braid links and Whitney rank polynomials of Lucas lattice

AlSukaiti,

Chbili

2024

Heliyon

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Alexander polynomials of alternating knots of genus two III

Cited by 10 publications

References 17 publications

On alternation numbers of links

On alternation numbers of links

On coxeter mapping classes and fibered alternating links

Alexander and Jones polynomials of weaving 3-braid links and Whitney rank polynomials of Lucas lattice

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