Knot Theory for Spatial Graphs Attached to a
                    Surface

Kawauchi, Akio

doi:10.1090/conm/670/13448

Cited by 5 publications

(3 citation statements)

References 16 publications

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“…Example 2.4. For n = 14, σ(a (1,7) ) is obtained as follows (see Figure 7): σ(a (1,7) ) = The components a (k,l) of M n satisfying 1 < k < 7 < l and a (k,l) = a (1,7) = 1 are a (2,8) , a (2,9) , a (2,10) , a (2,11) , a (2,12) , a (3,8) , a (3,9) , a (3,10) , a (3,11) , a (4,8) , a (4,9) , a (4,10) , a (5,8) , a (5,9) and a (6,8) .…”

Section: A Complete Graph Based On a Hamiltonian Cyclementioning

confidence: 99%

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A note on the cross-index of a complete graph based on a linear tree

Gokan

Katsumata

Nakajima

et al. 2018

J. Knot Theory Ramifications

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Section: A Complete Graph Based On a Hamiltonian Cyclementioning

confidence: 99%

“…σ(a (2,10) The components a (k,l) of M n satisfying 2 < k < 10 < l and a (k,l) = a (2,10) = 1 are a (3,11) , a (8,14) , a (9,13) and a (9,14) .…”

mentioning

confidence: 99%

A note on the cross-index of a complete graph based on a linear tree

Gokan

Katsumata

Nakajima

et al. 2018

J. Knot Theory Ramifications

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“…We mention here that a knotting probability of a spatial arc was defined directly from a knotting structure of a spatial graph but with the demerit that it depends on the heights of the crossing points of a diagram of the spatial arc in [3,5].…”

Section: Introductionmentioning

confidence: 99%

Unique diagram of a spatial arc and the knotting probability

Kawauchi

2019

Preprint

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It is shown that the projection image of an oriented spatial arc to any oriented plane is approximated by a unique arc diagram (up to isomorphic arc diagrams) determined from the spatial arc and the projection. In a separated paper, the knotting probability of an arc diagram is defined as an invariant under isomorphic arc diagrams. By combining them, the knotting probability of every oriented spatial arc is defined.

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The rank of a warping matrix

Akiyama

Shimizu

Watanabe

2016

Topology and its Applications

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The warping matrix has been defined for knot projections and knot diagrams by using warping degrees. In particular, the warping matrix of a knot diagram represents the knot diagram uniquely. In this paper we show that the rank of the warping matrix is one greater than the crossing number. We also discuss the linearly independence of knot diagrams by considering the warping incidence matrix.Theorem 1.1. Let P be an oriented knot projection on S 2 , and M(P ) the warping matrix of P . We have the following equality:where c(P ) is the crossing number of P .Let D be an oriented knot diagram on S 2 , and c(D) the crossing number of D. Let M (D) be the warping matrix of D without signs, which is mentioned concretely in Section 2. We also have the following theorem:The rest of this paper is organized as follows: In Section 2, we review the warping matrix. In Section 3, we prove Theorem 1.1 and 1.2. In Section 4, we define the warping incidence matrix of a knot diagram. In Section 5, we investigate linearly independence for knot diagrams.

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Knot Theory for Spatial Graphs Attached to a Surface

Abstract: Abstract. Beside a survey on several unknotting notions of a spatial graph done earlier by the author, unknotting notions on a spatial graph with degree one vertices attached to a surface are introduced.

Cited by 5 publications

References 16 publications

A note on the cross-index of a complete graph based on a linear tree

A note on the cross-index of a complete graph based on a linear tree

Unique diagram of a spatial arc and the knotting probability

The rank of a warping matrix

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