An application of non-positively curved cubings of alternating links

Sakuma, Masashi; Yokota, Yoshiyuki

doi:10.1090/proc/13918

Cited by 5 publications

(7 citation statements)

References 16 publications

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“…In the cases of (14)- (15), all endpoints of edges are adjacent or one of them is p, so we get the proof.…”

Section: Octahedral Triangulations Of Link Complementsmentioning

confidence: 77%

“…However, this deformation produces the problem of the existence of solutions because some triangulation constructed from a link diagram may have no solution. (A recent paper [15] proved the existence of solutions for the alternating links.) Furthermore, the author believes these deformation of the triangulation loses the combinatorial properties of link diagrams.…”

Section: Introductionmentioning

confidence: 99%

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Quandle theory and the optimistic limits of the representations of link groups

Cho

2018

Pacific J. Math.

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When a boudnary-parabolic representation of a link group to PSL(2,C) is given, Inoue and Kabaya suggested a combinatorial method to obtain the developing map of the representation using the octahedral triangulation and the shadow-coloring of certain quandle. Quandle is an algebraic system closely related with the Reidemeister moves, so their method changes quite naturally under the Reidemeister moves.In this article, we apply their method to the potential function, which was used to define the optimsitic limit, and construct a saddle point of the function. This construction works for any boundary-parabolic representation, and it shows that the octahedral triangulation is good enough to study all possible boundary-parabolic representations of the link group. Furthermore the evaluation of the potential function at the saddle point becomes the complex volume of the representation, and this saddle point changes naturally under the Reidemeister moves because it is constructed using the quandle.

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“…In the cases of (14)- (15), all endpoints of edges are adjacent or one of them is p, so we get the proof.…”

Section: Octahedral Triangulations Of Link Complementsmentioning

confidence: 77%

Section: Introductionmentioning

confidence: 99%

Quandle theory and the optimistic limits of the representations of link groups

Cho

2018

Pacific J. Math.

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“…In this section, we give an intuitive description of the Aitchison complexes and the Dehn complexes following [10,11]. For detailed description, see [9].…”

Section: And the Dehn Complexes For Alternating Linksmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…The Aitchison complex for an alternating link is actually a mapping cylinder of the natural map from the boundary of the exterior of the alternating link onto the Dehn complex. For a detailed description and historical background, see [9].In this paper, we introduce the concepts of a signed BW squared-complex (or an SBW squared-complex, for short) and a signed BW cubed-complex (or an SBW cubed-complex, for short), and give a combinatorial description of the Dehn complex and the Aitchison complex as an SBW squared-complex and an SBW cubedcomplex, respectively. The main theorem gives a necessary and sufficient condition for a given SBW cubed-complex to be isomorphic to the Aitchison complex of some alternating link exterior (Theorem 4.1).…”

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confidence: 99%

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A characterization of alternating link exteriors in terms of cubed complexes

Sakai

2018

J. Knot Theory Ramifications

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We give a characterization of alternating link exteriors in terms of cubed complexes. To this end, we introduce the concept of a "signed BW cubed-complex", and give a characterization for a signed BW cubed-complex to have the underlying space which is homeomorphic to an alternating link exterior. IntroductionRecently, Greene [5] and Howie [8], independently, established intrinsic characterizations of alternating links in terms of a pair of spanning surfaces, answering an old question of R. H. Fox. These results can be regarded as characterizations of alternating link exteriors which have marked meridians (see [8, Theorem 3.2]).The purpose of this paper is to give a characterization of alternating link exteriors from the viewpoint of cubed complexes. Our starting point is a cubical decomposition of alternating link exteriors, which is originally due to Aitchison, and is used by Agol [2], Adams [1], Thurston [10], Yokota [11,12] and Sakuma-Yokota [9]. Thus we call it the Aitchison complex. The Aitchison complex for an alternating link is actually a mapping cylinder of the natural map from the boundary of the exterior of the alternating link onto the Dehn complex. For a detailed description and historical background, see [9].In this paper, we introduce the concepts of a signed BW squared-complex (or an SBW squared-complex, for short) and a signed BW cubed-complex (or an SBW cubed-complex, for short), and give a combinatorial description of the Dehn complex and the Aitchison complex as an SBW squared-complex and an SBW cubedcomplex, respectively. The main theorem gives a necessary and sufficient condition for a given SBW cubed-complex to be isomorphic to the Aitchison complex of some alternating link exterior (Theorem 4.1). This implies a characterization of alternating link exteriors in terms of cubed complexes (Corollary 4.2). This paper is organized as follows. In Section 2, we give an intuitive description of the Aitchison complex and the Dehn complex following [10,11]. In Section 3, we introduce the SBW squared-complex and the SBW cubed-complex, and describe the Dehn complex and the Aitchison complex in terms of the SBW squared-complex and the SBW cubed-complex, respectively. In Section 4, we prove the main theorem.The author would like to thank his supervisor, Makoto Sakuma, for valuable suggestions. He would also like to thank Naoki Sakata and Takuya Katayama for their support and encouragement.2. An intuitive description of the Aitchison complexes arXiv:1804.03473v1 [math.GT]

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Idea of “Proof”

Murakami

Yokota

2018

Volume Conjecture for Knots

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An application of non-positively curved cubings of alternating links

Cited by 5 publications

References 16 publications

Quandle theory and the optimistic limits of the representations of link groups

Quandle theory and the optimistic limits of the representations of link groups

A characterization of alternating link exteriors in terms of cubed complexes

Idea of “Proof”

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