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Kouki Taniyama

Publications and source records attributed to Kouki Taniyama.

At least 19 recordsLinked to original sources
arXiv2024

Pairs of knot invariants

Let $α$ be a map from the set of all knot types ${\mathcal K}$ to a set $X$. Let $β$ be a map from ${\mathcal K}$ to a set $Y$. We define the relation between $α$ and $β$ to be the image of a map $(α,β)$ from ${\mathcal K}$ to $X\times Y$ sending an element $K$ of ${\mathcal K}$ to $(α(K),β(K))$. We determine the relations between $α$ and $β$ for certain $α$ and $β$ such as crossing number, unknotting number, bridge number, braid index, genus and canonical genus. This is a study of geography problem in knot theory.

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arXiv2022

Crossing numbers and rotation numbers of cycles in a plane immersed graph

For any generic immersion of a Petersen graph into a plane, the number of crossing points between two edges of distance one is odd. The sum of the crossing numbers of all $5$-cycles is odd. The sum of the rotation numbers of all $5$-cycles is even. We show analogous results for $6$-cycles, $8$-cycles and $9$-cycles. For any Legendrian spatial embedding of a Petersen graph, there exists a $5$-cycle that is not an unknot with maximal Thurston-Bennequin number, and the sum of all Thurston-Bennequin numbers of the cycles is $7$ times the sum of all Thurston-Bennequin numbers of the $5$-cycles. We show analogous results for a Heawood graph. We also show some other results for some graphs. We characterize abstract graphs that has a generic immersion into a plane whose all cycles have rotation number $0$.

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arXiv2020

Strong and weak (1, 3) homotopies on knot Projections

Strong and weak (1, 3) homotopies are equivalence relations on knot projections, defined by the first flat Reidemeister move and each of two different types of the third flat Reidemeister moves. In this paper, we introduce the cross chord number that is the minimal number of double points of chords of a chord diagram. Cross chord numbers induce a strong (1, 3) invariant. We show that Hanaki's trivializing number is a weak (1, 3) invariant. We give a complete classification of knot projections having trivializing number two up to the first flat Reidemeister moves using cross chord numbers and the positive resolutions of double points. Two knot projections with trivializing number two are both weak (1, 3) homotopy equivalent and strong (1, 3) homotopy equivalent if and only if they can be related by only the first flat Reidemeister moves. Finally, we determine the strong (1, 3) homotopy equivalence class containing the trivial knot projection and other classes of knot projections.

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arXiv2020

Circle arrangements of link projections

An oriented link projection is the image of a generic immersion of oriented circles into the 2-sphere. The circle arrangement of a link projection is a disjoint union of unoriented circles on the 2-sphere obtained by orientation-incoherent smoothing at each crossing point. We show that two oriented link projections have the same circle arrangement if and only if they are transformed into each other by certain local moves. We also show that every odd (resp. even) component link has a link projection whose circle arrangement consists of exactly one (resp. two) circles.

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arXiv2020

Unknotting numbers and crossing numbers of spatial embeddings of a planar graph

It is known that the unknotting number $u(L)$ of a link $L$ is less than or equal to half the crossing number $c(L)$ of $L$. We show that there are a planar graph $G$ and its spatial embedding $f$ such that the unknotting number $u(f)$ of $f$ is greater than half the crossing number $c(f)$ of $f$. We study relations between unknotting number and crossing number of spatial embedding of a planar graph in general.

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arXiv2020

Knot diagrams on a punctured sphere as a model of string figures

A string figure is topologically a trivial knot lying on an imaginary plane orthogonal to the fingers with some crossings. The fingers prevent cancellation of these crossings. As a mathematical model of string figure we consider a knot diagram on the $xy$-plane in $xyz$-space missing some straight lines parallel to the $z$-axis. These straight lines correspond to fingers. We study minimal number of crossings of these knot diagrams under Reidemeister moves missing these lines.

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arXiv2017

Achiral 1-cusped hyperbolic 3-manifolds not coming from amphicheiral null-homologous knot complements

It is experimentally known that achiral hyperbolic 3-manifolds are quite sporadic at least among those with small volume, while we can find plenty of them as amphicheiral knot complements in the 3-sphere. In this paper, we show that there exist infinitely many achiral 1-cusped hyperbolic 3-manifolds not homeomorphic to any amphicheiral null-homologous knot complement in any closed achiral 3-manifold.

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arXiv2017

Site-specific Gordian distances of spatial graphs

A site-specific Gordian distance between two spatial embeddings of an abstract graph is the minimal number of crossing changes from one to another where each crossing change is performed between two previously specified abstract edges of the graph. It is infinite in some cases. We determine the site-specific Gordian distance between two spatial embeddings of an abstract graph in certain cases. It has an application to puzzle ring problem. The site-specific Gordian distances between Milnor links and trivial links are determined. We use covering space theory for the proofs.

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arXiv2014

Mapping a knot by a continuous map

By a fixed continuous map from a $3$-space to itself, a knot in the $3$-space may be mapped to another knot in the $3$-space. We analyze possible knot types of them. Then we map a knot repeatedly by a fixed continuous map and analyze possible infinite sequences of knot types.

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arXiv2013

Discontinuous maps whose iterations are continuous

Let $X$ be a topological space and $f:X\to X$ a bijection. Let ${\mathcal C}(X,f)$ be a set of integers such that an integer $n$ is an element of ${\mathcal C}(X,f)$ if and only if the bijection $f^n:X\to X$ is continuous. A subset $S$ of the set of integers ${\mathbb Z}$ is said to be realizable if there is a topological space $X$ and a bijection $f:X\to X$ such that $S={\mathcal C}(X,f)$. A subset $S$ of ${\mathbb Z}$ containing 0 is called a submonoid of ${\mathbb Z}$ if the sum of any two elements of $S$ is also an element of $S$. We show that a subset $S$ of ${\mathbb Z}$ is realizable if and only if $S$ is a submonoid of ${\mathbb Z}$. Then we generalize this result to any submonoid in any group.

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arXiv2012

Multiplicity of a space over another space

We define a concept which we call multiplicity. First, multiplicity of a morphism is defined. Then the multiplicity of an object over another object is defined to be the minimum of the multiplicities of all morphisms from one to another. Based on this multiplicity, we define a pseudo distance on the class of objects. We define and study several multiplicities in the category of topological spaces and continuous maps, the category of groups and homomorphisms, the category of finitely generated $R$-modules and $R$-linear maps over a principal ideal domain $R$, and the neighbourhood category of oriented knots in the 3-sphere.

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arXiv2012

Plane curves in an immersed graph in $R^2$

For any chord diagram on a circle there exists a complete graph on sufficiently many vertices such that any generic immersion of it to the plane contains a plane closed curve whose chord diagram contains the given chord diagram as a sub-chord diagram. For any generic immersion of the complete graph on six vertices to the plane the sum of averaged invariants of all Hamiltonian plane curves in it is congruent to one quarter modulo one half.

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arXiv2011

$\triangle Y$-exchanges and the Conway-Gordon theorems

Conway-Gordon proved that for every spatial complete graph on 6 vertices, the sum of the linking numbers over all of the constituent 2-component links is congruent to 1 modulo 2, and for every spatial complete graph on 7 vertices, the sum of the Arf invariants over all of the Hamiltonian knots is also congruent to 1 modulo 2. In this paper, we give a Conway-Gordon type theorem for any graph which is obtained from the complete graph on 6 or 7 vertices by a finite sequence of $\triangle Y$-exchanges.

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arXiv2011

On intrinsically knotted or completely 3-linked graphs

We say that a graph is intrinsically knotted or completely 3-linked if every embedding of the graph into the 3-sphere contains a nontrivial knot or a 3-component link any of whose 2-component sublink is nonsplittable. We show that a graph obtained from the complete graph on seven vertices by a finite sequence of $\triangle Y$-exchanges and $Y \triangle$-exchanges is a minor-minimal intrinsically knotted or completely 3-linked graph.

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arXiv2009

Symmetries of spatial graphs and Simon invariants

An ordered and oriented 2-component link L in the 3-sphere is said to be achiral if it is ambient isotopic to its mirror image ignoring the orientation and ordering of the components. Kirk-Livingston showed that if L is achiral then the linking number of L is not congruent to 2 modulo 4. In this paper we study orientation-preserving or reversing symmetries of 2-component links, spatial complete graphs on 5 vertices and spatial complete bipartite graphs on 3+3 vertices in detail, and determine the necessary conditions on linking numbers and Simon invariants for such links and spatial graphs to be symmetric.

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arXiv2009

Almost positive links have negative signature

We analyze properties of links which have diagrams with a small number of negative crossings. We show that if a nontrivial link has a diagram with all crossings positive except possibly one, then the signature of the link is negative. If a link diagram has two negative crossings, we show that the signature of the link is nonpositive with the exception of the left-handed Hopf link (with possible trivial components). We also characterize those links which have signature zero and diagrams with two negative crossings. In particular, we show that if a nontrivial knot has a diagram with two negative crossings then the signature of the knot is negative, unless the knot is a twist knot with negative clasp. We completely determine all trivial link diagrams with two or fewer negative crossings. For a knot diagram with three negative crossings, the signature of the knot is nonpositive except the left-handed trefoil knot. These results generalize those of L. Rudolph, T. Cochran, E. Gompf, P. Traczyk, and J. H. Przytycki, solve Conjecture 5 of [P-2], and give a partial answer to Problem 2.8 of [Co-G] about knots dominating the trefoil knot or the trivial knot. We also describe all unknotting number one positive knots.

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arXiv2009

Braid presentation of spatial graphs

We define braid presentation of edge-oriented spatial graphs as a natural generalization of braid presentation of oriented links. We show that every spatial graph has a braid presentation. For an oriented link it is known that the braid index is equal to the minimal number of Seifert circles. We show that an analogy does not hold for spatial graphs.

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