arXiv subjects
Publications and source records attributed to Yuta Nozaki.
We show the homology cobordism group of homology $3$-spheres is not generated by graph homology $3$-spheres. Our technique is a combination of filtered instanton Floer theory and Gromov's simplicial volume.
A basic open question motivated by the study of ribbon knot diagrams asks whether every ribbon knot can be presented as a symmetric union. In this article, we give a negative answer to this question by exhibiting a ribbon Montesinos knot which does not admit a symmetric union presentation.
For a finitely generated group $G$, the Bieri-Neumann-Strebel-Renz (BNSR) invariants are subsets of the character sphere of $G$ that govern the finiteness properties of normal subgroups containing the commutator subgroup. We investigate the BNSR invariants of link groups and $2$-knot groups. In particular, for a link $L$ with at least two components, we prove that the commutator subgroup of the link group is finitely generated if and only if $L$ is a Hopf link. Moreover, we show that there exists a ribbon $2$-knot whose knot group has a non-symmetric BNS invariant.
For a link $L$ in the $3$-sphere, the $π$-orbifold group $G^\mathrm{orb}(L)$ is defined as a quotient of the link group $G(L)$ of $L$. When there exists an epimorphism $G^\mathrm{orb}(L)\to G^\mathrm{orb}(L')$ fitting into a certain commutative diagram, we define a relation $L\succeq L'$ and explore the relationships between the two links. Specifically, we prove that if $L\succeq L'$ and $L$ is a Montesinos link with $r$ rational tangles $(r\geq 3)$, then $L'$ is either a Montesinos link with at most $r+1$ rational tangles or a certain connected sum. We further show that if $L$ is a small link, then there are only finitely many links $L'$ satisfying $L\succeq L'$. In contrast, if $L$ has determinant zero, then $L\succeq L'$ for every $2$-bridge link $L'$. Our main applications concern symmetric unions of knots. In particular, we provide a criterion showing that a given knot does not admit a symmetric union presentation.
The Hopf index, a topological invariant that quantifies the linking of preimage fibers, is fundamental to the structure and stability of hopfions. In this work, we propose a new mathematical framework for modeling hopfions with high Hopf index, drawing on the language of singularity theory and the topology of differentiable maps. At the core of our approach is the notion of a generalized Hopf map of order $n$, whose structure is captured via fold maps and their Stein factorizations. We demonstrate that this theoretical construction not only aligns closely with recent experimental observations of high-Hopf-index hopfions, but also offers a precise classification of the possible configurations of fiber pairs associated to distinct points. Our results thus establish a robust bridge between the geometry of singular maps and the experimentally observed topology of complex field configurations of hopfions in materials and other physical systems.
For classical knots, Murasugi showed that the determinant modulo $8$ is classified by the Arf invariant. Boden and Karimi introduced a determinant for checkerboard colorable virtual knots. We prove that this determinant modulo $8$ is classified by the coefficient of $z^2$ in the ascending polynomial, an extension of the Conway polynomial for classical knots.
The Kontsevich invariant of links is independent of the choice of associator, whereas for tangles this is not the case in general. In this paper, we focus on $2$-component string links and investigate to what extent the Kontsevich invariant depends on the choice of associator. As an application, we show that the action of the unipotent part of the Grothendieck--Teichmüller group on the algebra of proalgebraic $2$-component string links is non-trivial, which provides a partial answer to a problem posed by Furusho.
Given finite simple graphs $G$ and $H$, the Hom complex $\mathrm{Hom}(G,H)$ is a polyhedral complex having the graph homomorphisms $G\to H$ as the vertices. We determine the homotopy type of each connected component of $\mathrm{Hom}(G,H)$ when $H$ is square-free, meaning that it does not contain the $4$-cycle graph $C_4$ as a subgraph. Specifically, for a connected $G$ and a square-free $H$, we show that each connected component of $\mathrm{Hom}(G,H)$ is homotopy equivalent to a wedge sum of circles. We further show that, given any graph homomorphism $f\colon G\to H$ to a square-free $H$, one can determine the homotopy type of the connected component of $\mathrm{Hom}(G,H)$ containing $f$ algorithmically.
For simple graphs $G$ and $H$, the Hom complex $\mathrm{Hom}(G,H)$ is a polyhedral complex whose vertices are the graph homomorphisms $G\to H$ and whose edges connect the pairs of homomorphisms which differ in a single vertex of $G$. Hom complexes play an important role in an algebro-topological approach to the graph coloring problem. It is known that $\mathrm{Hom}(G,H)$ is homotopy equivalent to a disjoint union of points and circles when both $G$ and $H$ are cycles. We generalize this known result by showing that the same holds whenever $G$ is connected and $H$ is a cycle. To this end, we explicitly construct the universal cover of each connected component of $\mathrm{Hom}(G,H)$ and prove that it is contractible. Additionally, we provide a simple criterion to determine whether the connected component containing a given homomorphism is homotopy equivalent to a point or circle.
In the study of ribbon knots, Lamm introduced symmetric unions inspired by earlier work of Kinoshita and Terasaka. We show an identity between the twisted Alexander polynomials of a symmetric union and its partial knot. As a corollary, we obtain an inequality concerning their genera. It is known that there exists an epimorphism between their knot groups, and thus our inequality provides a positive answer to an old problem of Jonathan Simon in this case. Our formula also offers a useful condition to constrain possible symmetric union presentations of a given ribbon knot. It is an open question whether every ribbon knot is a symmetric union.
Clasper surgery induces the $Y$-filtration $\{Y_n\mathcal{IC}\}_n$ over the monoid of homology cylinders, which serves as a $3$-dimensional analogue of the lower central series of the Torelli group of a surface. In this paper, we investigate the torsion submodules of the associated graded modules of these filtrations. To detect torsion elements, we introduce a homomorphism on $Y_n\mathcal{IC}/Y_{n+1}$ induced by the degree $n+2$ part of the LMO functor. Additionally, we provide a formula that computes this homomorphism under clasper surgery, and use it to demonstrate that every non-trivial torsion element in $Y_6\mathcal{IC}/Y_7$ has order $3$.
Youngs proved that every non-bipartite quadrangulation of the projective plane $\mathbb{R}\mathrm{P}^2$ is 4-chromatic. Kaiser and Stehl\'ık [J. Combin. Theory Ser. B 113 (2015), 1-17] generalised the notion of a quadrangulation to higher dimensions and extended Youngs' theorem by proving that every non-bipartite quadrangulation of the $d$-dimensional projective space $\mathbb{R}\mathrm{P}^d$ with $d \geq 2$ has chromatic number at least $d+2$. On the other hand, Hachimori et al. [European. J. Combin. 125 (2025), 104089] defined another kind of high-dimensional quadrangulation, called a normal quadrangulation. They proved that if a non-bipartite normal quadrangulation $G$ of $\mathbb{R}\mathrm{P}^d$ with any $d \geq 2$ satisfies a certain geometric condition, then $G$ is $4$-chromatic, and asked whether the geometric condition can be removed from the result. In this paper, we give a negative solution to their problem for the case $d=3$, proving that there exist 3-dimensional normal quadrangulations of $\mathbb{R}\mathrm{P}^3$ whose chromatic number is arbitrarily large. Moreover, we prove that no normal quadrangulation of $\mathbb{R}\mathrm{P}^d$ with any $d \geq 2$ has chromatic number $3$.
Nozaki et.~al.\ gave a homotopy classification of the knotted defects of ordered media in three-dimensional space by considering continuous maps from complements of spatial graphs to the order parameter space modulo a certain equivalence relation. We extend their result by giving a classification scheme for ordered media in handlebodies, where defects are allowed to reach the boundary. Through monodromies around meridional loops, global defects are described in terms of planar diagrams whose edges are colored by elements of the fundamental group of the order parameter space. We exhibit examples of this classification in octahedral frame fields and biaxial nematic liquid crystals.
Lord Kelvin proposed that atoms form hydrodynamic vortex knots. However, they typically untie through reconnections, i. e., local cut-and-slice events, unlike stable vortex unknots such as smoke rings. The same holds in superfluids--quantum fluids with zero viscosity--where vortices have quantized circulation, making them topologically stable. For over 150 years, hydrodynamically stable vortex knots have been sought both experimentally and theoretically. Here, we present the first demonstration of hydrodynamically stable vortex knots and links in experimentally realizable Bose-Einstein condensates of ultracold atomic gases and confirm it through dynamic simulations. Our method creates stable knotted vortex structures in systems where reconnections are prohibited, with potential relevance to neutron star interiors. Additionally, we anticipate our mathematical framework could have applications in quantum computation, quantum turbulence, and DNA dynamics, particularly where reconnections are restricted.
We give a homotopy classification of the global defects in ordered media, and explain it via the example of biaxial nematic liquid crystals, i.e., systems where the order parameter space is the quotient of the $3$-sphere $S^3$ by the quaternion group $Q$. As our mathematical model we consider continuous maps from complements of spatial graphs to the space $S^3/Q$ modulo a certain equivalence relation, and find that the equivalence classes are enumerated by the six subgroups of $Q$. Through monodromy around meridional loops, the edges of our spatial graphs are marked by conjugacy classes of $Q$; once we pass to planar diagrams, these labels can be refined to elements of $Q$ associated to each arc. The same classification scheme applies not only in the case of $Q$ but also to arbitrary groups.
A ladder lottery, known as ``Amidakuji'' in Japan, is a common way to decide an assignment at random. In this paper, we investigate reconfiguration and enumeration problems of cyclic ladder lotteries. First, when a permutation $π$ and an optimal displacement vector $\mathbf{x}$ are given, we investigate the reconfiguration and enumeration problems of the ``optimal'' cyclic ladder lotteries of $π$ and $\mathbf{x}$. Next, for a give permutation $π$ we consider reconfiguration and enumeration problems of the optimal displacement vectors of $π$.
We compute the Reidemeister-Turaev torsion of homology cylinders which takes values in the $K_1$-group of the $I$-adic completion of the group ring $\mathbb{Q}π_1Σ_{g,1}$, and prove that its reduction to $\widehat{\mathbb{Q}π_1Σ_{g,1}}/\hat{I}^{d+1}$ is a finite-type invariant of degree $d$. We also show that the $1$-loop part of the LMO homomorphism and the Enomoto-Satoh trace can be recovered from the leading term of our torsion.
We prove that the computation of a combinatorial shortest path between two vertices of a graph associahedron, introduced by Carr and Devadoss, is NP-hard. This resolves an open problem raised by Cardinal. A graph associahedron is a generalization of the well-known associahedron. The associahedron is obtained as the graph associahedron of a path. It is a tantalizing and important open problem in theoretical computer science whether the computation of a combinatorial shortest path between two vertices of the associahedron can be done in polynomial time, which is identical to the computation of the flip distance between two triangulations of a convex polygon, and the rotation distance between two rooted binary trees. Our result shows that a certain generalized approach to tackling this open problem is not promising. As a corollary of our theorem, we prove that the computation of a combinatorial shortest path between two vertices of a polymatroid base polytope cannot be done in polynomial time unless P = NP. Since a combinatorial shortest path on the matroid base polytope can be computed in polynomial time, our result reveals an unexpected contrast between matroids and polymatroids.