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Noboru Ito

Publications and source records attributed to Noboru Ito.

At least 19 recordsLinked to original sources
arXiv2026

Homological lifts of Arnold invariants $J^-$ and $J^+$

Viro's Euler-integral polynomial $P_C(q)$ and the Lanzat--Polyak quantized-curvature polynomial $I_q(C)$ refine Arnold's invariants $J^-$ and $J^+$ for generic immersed one-component plane curves. We construct homological lifts of both. The bigraded region homology retains the singular homology of every connected Alexander-index region; its graded Euler characteristic is $P_C(q)$. The triply graded smoothing-circle homology is generated by the oriented circles of the orientation-preserving smoothing and decategorifies to the smoothing term in $I_q(C)$. Keeping the actual region summands and the boundary regions of every smoothing circle gives a homological refinement of the oriented smoothing configuration, or Seifert state. An infinite family proves strictness: both polynomial data and the ordinary homological lifts agree, while the component-graded region homology and the branch-decomposed circle homology distinguish every pair. Further constructions recover the full $I_q(C)$ by a vertex complex, realize the local change of its curvature integral by edge homology, and give a canonical two-state homology for unoriented curves. Viro described his Euler-integral formula as an analogue of face state-sum formulas for quantum knot polynomials. Through the categorifications developed here, we obtain one concrete homological face-state-sum model realizing that analogy.

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arXiv2026

Commutators of pure twin groups and rondles

We formulate a relationship between finite-order rondle invariants with respect to triple-point modifications and the lower central series of subgroups of a pure twin group. Using our formulation, we construct infinitely many infinite sequences of prime plane curves such that, for each sequence, the curves share the same invariants up to any fixed order.

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arXiv2026

Underlying Stokes and de Rham structures for Arnold-type invariants

We introduce a framework on dual complexes for studying Arnold-type invariants of immersed curves and immersed surfaces via local finite-difference structures associated with Alexander numberings. For generic immersed plane curves and generic immersed surfaces, we define locally normalized maps $d^k \phi$ on dual skeleta and show that suitable evaluations recover the Arnold-type invariants $St_{(1)}$ and $St_{(2)}$. In particular, we establish normalized discrete Stokes-type compatibilities between adjacent dual skeleta and derive corresponding Shumakovitch-type identities for curves and surfaces. The normalization coefficients are determined by finite-difference factorial structures together with multiplicities of local configurations. We further interpret the iterated-integral-type structures appearing in Shumakovitch-type identities through finite-difference structures and highest-degree local Stokes compatibilities on dual complexes. We also reinterpret the slice formula for $St_{(2)}$ and $St_{(1)}$ as a compatibility relation between slicing and local operations on the dual complex. These results provide a unified framework in which global Arnold-type invariants arise as distribution-type evaluations of local data on dual complexes. The framework further clarifies the distinction between untwisted local closures and globally twisted structures such as the original Arnold invariant $St$, and suggests the existence of higher-degree local operations associated with the same dual-complex structure.

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arXiv2026

Minimal Generating Sets of Singular Reidemeister Moves and Their Classification

Singular knot theory extends classical knot theory by allowing transverse double points without over/under information, together with singular Reidemeister moves of types IV and V. A central open problem in this theory is to determine the minimal generating sets of oriented singular Reidemeister moves. In this paper, we completely solve this problem. In addition, we establish independence results for singular Reidemeister moves by introducing an invariant that provides obstructions and lower bounds for generating sets, including the independence of type III from types I, II, IV, and V. More precisely, starting from a minimal generating set of ordinary Reidemeister moves of types I--III, we prove that the singular moves admit exactly $96$ distinct inclusion-minimal generating sets, and that these exhaust all possibilities. Our proof introduces a new invariant for singular links, constructed via a projection to self-singular links, which detects the distinction between the two families of type IV moves and provides an obstruction for generating type V moves from types I--IV. We also determine the unoriented case, where the classification collapses to exactly $8$ minimal generating sets.

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arXiv2026

An explicit slice formula for surface invariants via curve invariants

We give an explicit slice formula for a surface invariant of generic immersions in $\mathbb{R}^3$, expressed in terms of curve invariants arising from planar slices. Using a motion-picture viewpoint, we introduce differential measures that record local changes of the curve invariant $St_{(1)}$ and the surface invariant $St_{(2)}$ across singular slice transitions. Our main result shows that, for a quadruple-point event, if $j$ denotes the number of outward coorientations before the event, then the change of the surface invariant satisfies $dSt_{(2)} = 2j - 4$. This yields a computable and combinatorial description of the surface invariant via slice data. In particular, the formula makes explicit the relation between curve-level invariants and finite-order invariants of surface immersions in the sense of Nowik.

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arXiv2026

B-type coefficient polynomial

An A-type coefficient polynomial introduced by Kawauchi recovers the HOMFLY-PT polynomial as a formal power series within skein theory. A notable feature of this construction is that each coefficient defines a link invariant, yielding an infinite sequence of invariants, while the low-degree coefficients are relatively easy to compute. In this paper, we extend this viewpoint to the B-type setting. Unlike the A-type case, the B-type setting requires a genuinely new inductive scheme due to the four-term skein relation. More precisely, we introduce coefficient polynomials associated with the B-type skein relation and show that their generating series recovers the Kauffman polynomial. We further prove that these coefficient polynomials are well-defined and that the resulting generating series is invariant under the corresponding Reidemeister moves.

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arXiv2026

Genera of two-component alternating links

We extend the equality-type results of Ito--Takimura and Kindred for the non-orientable genera of alternating knots to the setting of two-component alternating links. We show that, for such links, a unified quantity capturing both orientable and non-orientable genera is completely determined by the splice sequence realizing the splice-unknotting number up to an explicit correction term.

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arXiv2025

Minimal Generating sets of Reidemeister moves

We determine whether each known generating set of arbitrary oriented Reidemeister moves is minimal. We then provide a complete classification of minimal generating sets that include a coherent Reidemeister move of type II. We also classify all minimal generating sets that include a braid-type Reidemeister move of type III. Beyond these two cases, we identify 16 possible candidates for minimal generating sets. Among them, we prove that 12 are indeed minimal, whereas the minimality and even the generating property of the remaining 4 sets remains unsolved (Remark 5.1).

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arXiv2025

Remark on twists of Frobenius algebra and link homology

We discuss twists on Frobenius algebras in the context of link homology. In his paper in 2006, Khovanov asserted that a twist of a Frobenius algebra yields an isomorphic chain complex on each link diagram. Although the result has been widely accepted for nearly two decades, a subtle gap in the original proof was found in the induction step of the construction of the isomorphism. Following discussion with Khovanov, we decided to provide a new proof. Our proof is based on a detailed analysis of configurations of circles in each state.

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arXiv2025

Arnold Strangeness of surface immersions

It is known that for any smooth sphere eversion, the number of quadruple point jumps is always odd. In this paper, we define an integer-valued function that detects and classifies jumps involving quadruple points and triple-line tangencies. Our function provides a higher-dimensional analogue of the Arnold strangeness invariant for plane curves. It classifies quadruple point jumps into the five geometrically distinct cases based on coorientation data and reflects finer geometric features for generic immersions of closed surfaces into the 3-space.

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arXiv2024

Lifting link invariants by functors on nanophrases

Nanophrases have a filtered structure consisting of an infinite number of categories, and each category has a homotopy structure. Among these categories, the one that we are most familiar with is the category of links. Interestingly, the category in which the Jones polynomial is defined consists of a category that is actually less informative category than the link category, and so is the quandle category. The former is called the pseudolink category and the latter the quasilink category, which are covered by the link category; there is a known filtration: links cover pseudolinks, pseudolinks cover virtual strings, and virtual strings cover free links. The introduction and groundwork for nanophrases were done by Turaev around 2005. In this paper, we introduce functors (Theorem 1) from general nanophrases to virtual strings/pseudolinks/quasilinks/free links. These functors are powerful because of the difference between each of two. To demonstrate the effectiveness of such new construction of functors, the Jones pseudolink polynomial (Section 4), which implies the Jones link polynomial, is extended to general filtered nanophrases using one of the functors (Section 5). In particular, the information level of the nanophrases used in each category is explicitly described in the above construction process. This paper is a refined version by developing parts of sections 3 and 6 in arXiv: 0901.3956.

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arXiv2023

Automatic computation of crosscap number of alternating knots

We specify the computational complexity of crosscap numbers of alternating knots by introducing an automatic computation. For an alternating knot $K$, let $\cal{E}$ be the number of edges of its diagram. Then there exists a code such that the complexity of this computation of the crosscap number of $K$ is estimated by $O(\cal{E}^3)$.

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arXiv2022

Multiple Linking Number

The linking number is the simplest link invariant given by Gauss; it is the first Gauss diagram formula expressed by one arrow among two circles. Proceeding the next stage, we study the second Gauss diagram formula consisting of two arrows among two circles. We call a function of this type the multiple linking number. There are two multiple linking numbers; one of them is ordinary Vassiliev invariant and the other function is surprisingly sensitive to the necessity of the second Reidemeister moves though any one-component Gauss diagram formula cannot detect the necessity.

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arXiv2022

Curvature and quantized Arnold strangeness

By integrating curvatures multiplied non-trivial densities, we introduce an integral expression of the Arnold strangeness that is a celebrated plane curve invariant. The key is a partition function by Shumakovitch to reformulate Arnold strangeness. Our integrating curvatures suggests a quantized Arnold strangeness which Taylor expansion includes the rotation number and the original Arnold strangeness, and also higher terms are invariants of Tabachnikov. It is an analogue of the quantization by Viro for Arnold $J^-$ and by Lanzat-Polyak for $J^+$.

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arXiv2022

Variations of Milnor's triple linking number

Topological polymers have various topological types, and they are expressed by graphs. However, the Jones polynomial, we have a difficulty to compute it; computational time is growing exponentially with respect to the crossing number. The simplest Vassiliev invariant is the linking number and thus we will seek a next simple one is as the Milnor's triple linking number. In this paper, we introduce simple Gauss diagram formulas of Vassiliev invariants of Milnor type. These are non-torsion valued, whereas the base-point-free Milnor's triple linking number is usually torsion-valued.

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arXiv2022

A triple coproduct of curves and knots

We introduce a triple coproduct for knots on surfaces, providing a commutative framework that decomposes a single-component diagram into three components (Section 2). This construction is motivated by the interplay between intersection theory and the affine index polynomial, and extends these ideas to a three-component setting (Section 5). Building on Turaev's cobracket theory, we define an integer-valued invariant under stable equivalence by combining the coproduct with an intersection-theoretic function (Theorem 1). Unlike classical cobrackets, which often collapse distinct local configurations, our approach preserves combinatorial traces of smoothing choices, enabling fine-grained detection of local crossing patterns (Definition 4). In the symmetric tensor setting, Reidemeister invariance uniquely determines the relations in the word space (Equations (4), (5)) and canonically fixes smoothing weights, revealing an intrinsic simplicity behind the algebraic framework (Corollary 1). This uniqueness result positions our construction as the canonical commutative analogue of Turaev's non-commutative cobracket and clarifies its interpretation as a classical limit of skein quantization, extending the theoretical scope beyond previously known invariants (Section 6). Examples demonstrate substantial distinguishing power, separating an infinite sequence of knots arising from distinct smoothing choices and broadening the reach of existing invariants (Proposition 2).

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arXiv2022

$\mathfrak{gl}(1 \vert 1)$-Alexander polynomial for $3$-manifolds

As an extension of Reshetikhin and Turaev's invariant, Costantino, Geer and Patureau-Mirand constructed $3$-manifold invariants in the setting of relative $G$-modular categories, which include both semisimple and non-semisimple ribbon tensor categories as examples. In this paper, we follow their method to construct a $3$-manifold invariant from Viro's $\mathfrak{gl}(1\vert 1)$-Alexander polynomial. We take lens spaces $L(7, 1)$ and $L(7, 2)$ as examples to show that this invariant can distinguish homotopy equivalent manifolds.

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