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Noboru Ito

Publications and source records attributed to Noboru Ito.

At least 55 records · Page 3Linked to original sources
arXiv2020

A cobordism realizing crossing change on $\mathfrak{sl}_2$ tangle homology and a categorified Vassiliev skein relation

In this paper, we discuss degree 0 crossing change on Khovanov homology in terms of cobordisms. Namely, using Bar-Natan's formalism of Khovanov homology, we introduce a sum of cobordisms that yields a morphism on complexes of two diagrams of crossing change, which we call the "genus-one morphism." It is proved that the morphism is invariant under the moves of double points in tangle diagrams. As a consequence, in the spirit of Vassiliev theory, taking iterated mapping cones, we obtain an invariant for singular tangles that extending sl(2) tangle homology; examples include Lee homology, Bar-Natan homology, and Naot's universal Khovanov homology as well as Khovanov homology with arbitrary coefficients. We also verify that the invariant satisfies categorified analogues of Vassiliev skein relation and the FI relation.

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arXiv2020

Gauss diagram formulas of Vassiliev invariants of spatial 2-bouquet graphs

We introduce new formulas that are Vassiliev invariants of flat vertex isotopy classes of spatial 2-bouquet graphs, which are equivalent to 2-string links. Although any Gauss diagram formula of Vassiliev invariants of spatial 2-bouquet graphs in a 3-space has been unknown, this paper gives the first and simple example.

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arXiv2020

Knot projections with reductivity two

Reductivity of knot projections refers to the minimum number of splices of double points needed to obtain reducible knot projections. Considering the type and method of splicing (Seifert type splice or non-Seifert type splice, recursively or simultaneously), we can obtain four reductivities containing Shimizu's reductivity, three of which are new. In this paper, we determine knot projections with reductivity two for all four of the definitions. We also provide easily calculated lower bounds for some reductivities. Further, we detail properties of each reductivity, and describe relationships among the four reductivities with examples.

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arXiv2020

Survey and remarks on Viro's definition of Khovanov homology

This paper reviews and offers remarks upon Viro's definition of the Khovanov homology of the Kauffman bracket of unoriented framed tangles (Sec. 2). The review is based on a file of his talk. This definition contains an exposition of the relation between the $R$-matrix and the Kauffman bracket (Sec. 2.2).

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arXiv2020

Triple chords and strong (1, 2) homotopy

A triple chord is a sub-diagram of a chord diagram that consists of a circle and finitely many chords connecting the preimages for every double point on a spherical curve, and it has exactly three chords giving the triple intersection. This paper describes some relationships between the number of triple chords and an equivalence relation called strong (1, 2) homotopy, which consists of the first and one kind of the second Reidemeister moves involving inverse self-tangency if the curve is given any orientation. We show that a prime knot projection is trivialized by strong (1, 2) homotopy, if it is a simple closed curve or a prime knot projection without 1- and 2-gons whose chord diagram does not contain any triple chords. We also discuss the relation between Shimizu's reductivity and triple chords.

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arXiv2020

Strong and weak (1, 2, 3) homotopies on knot projections

A knot projection is an image of a generic immersion from a circle into a two-dimensional sphere. We can find homotopies between any two knot projections by local replacements of knot projections of three types, called Reidemeister moves. This paper defines an equivalence relation for knot projections called weak (1, 2, 3) homotopy, which consists of Reidemeister moves of type 1, weak type 2, and weak type 3. This paper defines the first non-trivial invariant under weak (1, 2, 3) homotopy. We use this invariant to show that there exist an infinite number of weak (1, 2, 3) homotopy equivalence classes of knot projections. By contrast, all equivalence classes of knot projections consisting of the other variants of a triple type, i.e., Reidemeister moves of (1, strong type 2, strong type 3), (1, weak type 2, strong type 3), and (1, strong type 2, weak type 3), are contractible.

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arXiv2020

Sub-chord diagrams of knot projections

A chord diagram is a circle with paired points with each pair of points connected by a chord. Every generic immersed spherical curve provides a chord diagram by associating each chord with two preimages of a double point. Any two spherical curves can be related by a finite sequence of three types of local replacement RI, RII, and RIII, called Reidemeister moves. This study counts the difference in the numbers of sub-chord diagrams embedded in a full chord diagram of any spherical curve by applying one of the moves RI, strong RII, weak RII, strong RIII, and weak RIII defined by connections of branches related to the local replacements (Theorem 1). This yields a new integer-valued invariant under RI and strong RIII that provides a complete classification of prime reduced spherical curves with up to at least seven double points (Theorem 2, Fig. 24): there has been no such invariant before. The invariant expresses the necessary and sufficient condition that spherical curves can be related to a simple closed curve by a finite sequence of RI and strong RIII moves (Theorem 3). Moreover, invariants of spherical curves under flypes are provided by counting sub-chord diagrams (Theorem 4).

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arXiv2020

Higher-order finite type invariants of classical and virtual knots and unknotting operations

Vassiliev introduced filtered invariants of knots using an unknotting operation, called crossing changes. Goussarov, Polyak, and Viro introduced other filtered invariants of virtual knots, which order is called GPV-order, using an unknotting operation, called virtualization. We defined other filtered invariants, which order is called $F$-order, of virtual knots using an unknotting operation, called forbidden moves. In this paper, we show that the set of virtual knot invariants of $F$-order $\le n+1$ is strictly stronger than that of $F$-order $\le n$ and that of GPV-order $\le 2n+1$. To obtain the result, we show that the set of virtual knot invariants of $F$-order $\le n$ contains every Goussarov-Polyak-Viro invariant of GPV-order $\le 2n+1$, which implies that the set of virtual knot invariants of $F$-order is a complete invariant of classical and virtual knots.

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arXiv2020

Based chord diagrams of spherical curves

This paper demonstrates an approach for developing a framework to produce invariants of base-point-free generic spherical curves under some chosen local moves from Reidemeister moves using based chord diagrams. Our invariants not only contain Arnold's classical generic spherical curve invariant but also new invariants.

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arXiv2020

A colored Khovanov bicomplex

In this note, we prove the existence of a tri-graded Khovanov-type bicomplex (Theorem 1.2). The graded Euler characteristic of the total complex associated with this bicomplex is the colored Jones polynomial of a link. The first grading of the bicomplex is a homological one derived from cabling of the link (i.e., replacing a strand of the link with several parallel strands); the second grading is related to the homological grading of ordinary Khovanov homology; finally, the third grading is preserved by the differentials, and corresponds to the degree of the variable in the colored Jones polynomial. In particular, we introduce a way to take a small cabling link diagram directly from a big cabling link diagram (Theorem 3.2). This is a refined version of arXiv:0907.5247.

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arXiv2020

(1, 2) and weak (1, 3) homotopies on knot projections

In this paper, we obtain the necessary and sufficient condition that two knot projections are related by a finite sequence of the first and second flat Reidemeister moves (Theorem 1). We also consider an equivalence relation that is called weak (1, 3) homotopy. This equivalence relation occurs by the first flat Reidemeister move and one of the third flat Reidemeister moves. We introduce a map sending weak (1, 3) homotopy classes to knot isotopy classes (Sec. 3). Using the map, we determine which knot projections are trivialized under weak (1, 3) homotopy (Corollary 3).

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arXiv2020

On Khovanov complexes

In this paper, we discuss a proof of the isotopy invariance of a parametrized Khovanov link homology including categorifications of the Jones polynomial and the Kauffman bracket polynomial though it is a known fact. In order to present a proof easy-to-follow, we give an explicit description of retractions and chain homotopies between complexes to induce the invariance under isotopy of links. This is a refined version of arXiv: 0907.2104.

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arXiv2020

New deformations on spherical curves and Östlund Conjecture

In 2018, Funakoshi, Hashizume, Ito, Kobayashi, and Murai used a deformation of spherical curves called deformation type $α$. Then, it was showed that if two spherical curves $P$ and $P'$ are equivalent under the relation consisting of deformations of type RI and type RIII up to ambient isotopy, and satisfy certain conditions, then $P'$ is obtained from $P$ by a finite sequence of deformations of type $α$. In this paper, we introduce a new type of deformations of spherical curves, called deformation of type $β$. The main result of this paper is: Two spherical curves $P$ and $P'$ are equivalent under (possibly empty) deformations of type RI and a single deformation of type RIII up to ambient isotopy if and only if reduced(P) and reduced(P') are transformed each other by exactly one deformation which is of type RIII, type $α$, or type $β$ up to ambient isotopy, where reduced(Q) is the spherical curve which does not contain a $1$-gon obtained from a spherical curve $Q$ by applying deformations of type RI up to ambient isotopy.

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arXiv2019

Crossing change on Khovanov homology and a categorified Vassiliev skein relation

Khovanov homology is a categorification of the Jones polynomial, so it may be seen as a kind of quantum invariant of knots and links. Although polynomial quantum invariants are deeply involved with Vassiliev (aka. finite type) invariants, the relation remains unclear in case of Khovanov homology. Aiming at it, in this paper, we discuss a categorified version of Vassiliev skein relation on Khovanov homology. More precisely, we will show that the "genus-one" operation gives rise to a crossing change on Khovanov complexes. Invariance under Reidemeister moves turns out, and it enables us to extend Khovanov homology to singular links. We then see that a long exact sequence of Khovanov homology groups categorifies Vassiliev skein relation for the Jones polynomials. In particular, the Jones polynomial is recovered even for singular links. We in addition discuss the FI relation on Khovanov homology.

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arXiv2019

Arrow diagrams on spherical curves and computations

We give a definition of an integer-valued function $\sum_i α_i x ^*_i$ derived from arrow diagrams for the ambient isotopy classes of oriented spherical curves. Then, we introduce certain elements of the free $\mathbb{Z}$-module generated by the arrow diagrams with at most $l $ arrows, called relators of Type~($\check{\rm{I}}$) (($\check{\rm{SI\!I} }$), ($\check{\rm{WI\!I}}$), ($\check{\rm{SI\!I\!I}}$), or ($\check{\rm{ WI\!I\!I}}$), resp.), and introduce another function $\sum_i α_i \tilde{x}^*_i$ to obtain $\sum_i α_i x^*_i$. One of the main results shows that if $\sum_i α_i \tilde{x}^*_i$ vanishes on finitely many relators of Type~($\check{\rm{I}}$) (($\check{\rm{SI\!I}}$) , ($\check{\rm{WI\!I}}$), ($\check{\rm{SI\!I\!I}}$), or ($\check{\rm{WI\! I\!I}}$), resp.), then $\sum_i α_i \tilde{x}$ is invariant under the deformation of type $\rm{RI}$ (strong$\rm{RI\!I}$, weak$\rm{RI\!I}$, strong$\rm{RI\!I\!I}$, or weak$\rm{RI\!I\!I}$, resp.). The other main result is that we obtain functions of arrow diagrams with up to six arrows. This computation is done with the aid of computers.

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arXiv2019

On a Poincaré polynomial from Khovanov homology and Vassiliev invariants

We introduce a Poincaré polynomial with two-variable $t$ and $x$ for knots, derived from Khovanov homology, where the specialization $(t, x)$ $=$ $(1, -1)$ is a Vassiliev invariant of order $n$. Since for every $n$, there exist non-trivial knots with the same value of the Vassiliev invariant of order $n$ as that of the unknot, there has been no explicit formulation of a perturbative knot invariant which is a coefficient of $y^n$ by the replacement $q=e^y$ for the quantum parameter $q$ of a quantum knot invariant, and which distinguishes the above knots together with the unknot. The first formulation is our polynomial.

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arXiv2017

Spectral sequences of colored Jones polynomials, colored Rasmussen invariants and nanophrases

We introduce three spectral sequences which give some expressions of colored Jones polynomials. Each spectral sequence contains a Khovanov-type homology groups. Two of them are derived from a bicomplex of the colored Jones polynomial. The other is the spectral sequence that deduces a colored Rasmussen invariant of links. We also introduce three functors between categories of nanophrases, generalizations of links, and obtain their applications using colored Jones polynomials and their categorifications.

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