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Masakazu Teragaito

Publications and source records attributed to Masakazu Teragaito.

At least 19 recordsLinked to original sources
arXiv2026

Asymmetric L-space knots with arbitrary braid index

The first examples of asymmetric L-space knots were found by Baker and Luecke. Among Baker-Luecke knots, the simplest one has braid index 12. Later, it turned out that there are just 9 asymmetric hyperbolic L-space knots in the SnapPy census, and their braid indices take the values 4, 5, 6 and 7. It is known that asymmetric L-space knots have braid index at least 4. Recently, Baker and the second author give infinitely many asymmetric hyperbolic L-space knots with braid index 4. In this paper, we construct an asymmetric hyperbolic L-space knot with arbitrary braid index bigger than 4. In fact, there exist infinitely many such knots for each braid index.

math.GT↗
arXiv2026

Every non-trivial knot group is fully residually perfect

Given a class $\mathcal{P}$ of groups we say that a group $G$ is fully residually $\mathcal{P}$ if for any finite subset $F$ of $G$, there exists an epimorphism from $G$ to a group in $\mathcal{P}$ which is injective on $F$. It is known that any non-trivial knot group is fully residually finite. For hyperbolic knots, its knot group is fully residually closed hyperbolic $3$--manifold group, and fully residually simple. In this article, we show that every non-trivial knot group is fully residually perfect, closed $3$--manifold group.

math.GT↗
arXiv2026

On hyperbolic L-space knots with braid index four and tunnel number two

There are only three known strongly invertible hyperbolic L-space knots with braid index four and tunnel number two. They are t09284, t10496 and o9_34409 in the SnapPy census. In this paper, we give the first infinite family of strongly invertible hyperbolic L-space knots with braid index four and tunnel number two that includes t10496 and o9_34409.

math.GT↗
arXiv2026

Dehn filling and the knot group II: Ubiquity of persistent elements

Let $K$ be a nontrivial knot in $S^3$. We say that an element of the knot group $G(K)$ is \textit{persistent} if it remains nontrivial under all nontrivial Dehn fillings. Such elements exist for every nontrivial knot. Indeed, Property P is equivalent to the statement that the meridian of $K$ is a persistent element, and this represents the first instance of such elements. Building on the solution to the Property P conjecture due to Kronheimer and Mrowka, we show that every nontrivial knot group admits infinitely many persistent elements with pairwise disjoint automorphic orbits, none of which contains a power of the meridian. We then develop this further to show that for a broad class of hyperbolic knots - namely those admitting no surgery whose resulting manifold has torsion in its fundamental group - persistent elements are not rare curiosities, but rather structurally pervasive in $G(K)$. This is reflected in the following two properties: (i) Every subgroup of $G(K)$ that is not contained in the normal closure of a peripheral element contains persistent elements. (ii) Persistent elements exist outside every proper subgroup of $G(K)$.

math.GT↗
arXiv2026

An exploration of low crossing and chiral cosmetic bands with grid diagrams

We computationally explore non-coherent band attachments between low crossing number knots, using grid diagrams. We significantly improve the current H(2)-distance table. In particular, we find two new distance one pairs with fewer than seven crossings: one between $3_1\#3_1$ and $7_4m$, and a chirally cosmetic one for $7_3$. We further determine a total of 33 previously unknown H(2)-distance one pairs for knots with up to $8$ crossings. The appendix by Kazuhiro Ichihara, In Dae Jong and Masakazu Teragaito contains a construction explaining the existence of chirally cosmetic bands for an infinite family of knots, including $5_1,\, 7_3$ and $8_8$.

math.GT↗
arXiv2026

Group theoretic perspective on Dehn fillings: Property P conjecture and beyond

The Property P Conjecture, which was settled by Kronheimer and Mrowka, asserts that every $3$--manifold obtained by non-trivial Dehn surgery on a non-trivial knot is never simply connected. We propose new perspectives in studying Dehn filling from group theoretic point of view, which stem from several variation of the Property P conjecture.

math.GT↗
arXiv2025

On asymmetric hyperbolic L-space knots of braid index four

A knot is called an L-space knot if it admits a positive Dehn surgery yielding an L-space. In the SnapPy census, there are exactly 9 asymmetric L-space knots. Among them, the knot t12533 is the only known example of braid index 4. We generalize this knot, and give the first infinite family of asymmetric hyperbolic L-space knots of braid index 4.

math.GT↗
arXiv2025

Dehn filling and the knot group I: Realization Property

Each $r$-Dehn filling of the exterior $E(K)$ of a knot $K$ in $S^3$ produces a $3$-manifold $K(r)$, and induces an epimorphism from the knot group $G(K) = π_1(E(K))$ to $π_1(K(r))$, which trivializes elements in its kernel. To each element $g \in G(K)$, consider all the non-trivial Dehn fillings and assign $\mathcal{S}_K(g) = \{ r \in \mathbb{Q} \mid \textrm{$r$-Dehn filling trivializes}\ g \}$ $\subset \mathbb{Q}$. Which subsets of $\mathbb{Q}$ can occur as $\mathcal{S}_K(g)$? Property P concerns this question and gives a fundamental result which asserts that the emptyset can be realized by $\mathcal{S}_K(μ)$ for the meridian $μ$ of $K$. Suppose that $K$ is a hyperbolic knot. Then $\mathcal{S}_K(g)$ is known to be finite for all non-trivial elements $g \in G(K)$. We prove that generically, for instance, if $K$ has no exceptional surgery, then any finite (possibly empty) family of slopes $\mathcal{R} = \{ r_1, . . . , r_n \}$ can be realized by $\mathcal{S}_K(g)$ for some element $g \in G(K)$. Furthermore, there are infinitely many, mutually non-conjugate such elements, each of which is not conjugate to any power of $g$. We also provide an example showing that the above realization property does not hold unconditionally.

math.GT↗
arXiv2024

Hyperbolic knots with arbitrarily large torsion order in knot Floer homology

In knot Floer homology, there are two types of torsion order. One is the minimal power of the action of the variable $U$ to annihilate the $\mathbb{F}_2[U]$-torsion submodule of the minus version of knot Floer homology $\mathrm{HFK}^-(K)$. This is introduced by Juhász, Miller and Zemke, and denoted by $\mathrm{Ord}(K)$. The other, $\mathrm{Ord}'(K)$, introduced by Gong and Marengon, is similarly defined for the $\mathbb{F}_2[U]$-torsion submodule of the unoriented knot Floer homology $\mathrm{HFK}'(K)$. For both torsion orders, it is known that arbitrarily large values are realized by torus knots. In this paper, we prove that they can be realized by hyperbolic knots, most of which are twisted torus knots. Two torsion orders are argued in a unified way by using the Upsilon torsion function introduced by Allen and Livingston. We also give the first infinite family of hyperbolic knots which shares a common Upsilon torsion function.

math.GT↗
arXiv2024

Twisted right-angled Artin groups embedded in knot groups

Twisted right-angled Artin groups are defined through presentation based on mixed graphs. Each vertex corresponds to a generator, each undirected edge yields a commuting relation and each directed edge gives a Klein bottle relation. If there is no directed edge, then this reduces to an ordinary right-angled Artin group. There is a characterization of right-angled Artin groups which can be embedded in knot groups by Katayama. In this paper, we completely determine twisted right-angled Artin groups embedded in knot groups.

math.GT↗
arXiv2024

Homotopy classification of knotted defects in ordered media

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.

cond-mat.soft↗
arXiv2024

Classification of generalized torsion elements of order two in 3-manifold groups

Let $G$ be a group and $g$ a non-trivial element in $G$. If some non-empty finite product of conjugates of $g$ equals to the identity, then $g$ is called a generalized torsion element. The minimum number of conjugates in such a product is called the order of $g$. We will classify $3$-manifolds $M$, each of whose fundamental group has a generalized torsion element of order two. Furthermore, we will classify such elements in $π_1(M)$. We also prove that $R$-group and $\overline{R}$-group coincide for $3$-manifold groups, and classify $3$-manifold groups which are $R$-groups (and hence $\overline{R}$-groups).

math.GT↗
arXiv2024

Hyperbolic L-space knots and their Upsilon invariants

For a knot in the 3-sphere, the Upsilon invariant is a piecewise linear function defined on the interval [0,2]. For an L-space knot, the Upsilon invariant is determined only by the Alexander polynomial of the knot. We exhibit infinitely many pairs of hyperbolic L-space knots such that two knots of each pair have distinct Alexander polynomials, so they are not concordant, but share the same Upsilon invariant. Conversely, we examine the restorability of the Alexander polynomial of an L-space knot from the Upsilon invariant through the Legendre-Fenchel transformation.

math.GT↗
arXiv2022

Hyperbolic L-space knots and their formal semigroups

For an L-space knot, the formal semigroup is defined from its Alexander polynomial. It is not necessarily a semigroup. That is, it may not be closed under addition. There exists an infinite family of hyperbolic L-space knots whose formal semigroups are semigroups generated by three elements. In this paper, we give the first infinite family of hyperbolic L-space knots whose formal semigroups are semigroups generated by five elements.

math.GT↗