arXiv subjects
Publications and source records attributed to Tetsuya Ito.
We give a lower bound of the minimum crossing number of knots and links projected on a 2-sided surface in a 3-manifold using the rank of fundamental groups which means that the crossing number can take arbitrary large values. On the contrary, we show that for a branched surface, the minimum crossing number is always zero. As an application, we give an upper bound of the three-page index by the crossing number.
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 $G(K)$.
The 3-dimensional clasp number $cl(K)$ of a knot $K$ is the minimum number of clasp singularities of clasp disk, a singular immersed disk bounding $K$ whose singular set consists of only clasp singularities. We give a classification of clasp number two, genus two fibered knots under the assumption that they admit a clasp disk of certain type which we call of type II.
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.
For a non-split multi-crossing diagram $D$ of a link $L$ we show that $\alpha(L)-2 \leq c_2(D) + \sum_{n> 2}(2n-4)c_n(D)$ holds. Here $\alpha(L)$ is the arc index and $c_n(D)$ is the number of $n$-crossings of $D$. This generalizes and subsumes many known inequalities related to multi-crossing numbers. In the course of proof, we introduce a notion of weak rectangular diagram and show that a loose rectangular diagram can be converted to usual rectangular diagram preserving its arc index.
We study the negative band number of braids, knots, and links using Birman, Ko, and Lee's left-canonical form of a braid. As applications, we characterize up to conjugacy strongly quasipositive braids and almost strongly quasipositive braids.
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.
Cromwell proved that the minimum $v$-degree of the HOMFLY polynomial of homogeneous link $L$ is bounded above by $1-\chi(L)$, where $\chi(L)$ is the maximum Euler characteristic of Seifert surfaces of $L$. We prove its slice version, stating that the minimum $v$-degree of the HOMFLY polynomial of homogeneous link $L$ is bounded above by $1-\chi_4(L)$, the maximum 4-dimensional Euler characteristic of $L$. As a byproduct, we prove a conjecture of Stoimenow that for an alternating link, the minimum $v$-degree of the HOMFLY polynomial is smaller than or equal to its signature.
The Ma-Qiu index of a group is the minimum number of normal generators of the commutator subgroup. We show that the Ma-Qiu index gives a lower bound of the presentation distance of two groups, the minimum number of relator replacements to change one group to the other. Since many local moves in knot theory induce relator replacements in knot groups, this shows that the Ma-Qiu index of knot groups gives a lower bound of the Gordian distance based on various local moves. In particular, this gives a unified and simple proof of the Nakanishi index bounds of various unknotting numbers, including virtual or welded knot cases.
For a given knot $K$ and $w>0$, we construct infinitely many mutually distinct hyperbolic knots $\{K_i\}$ such that the $P$-satellites of $K$ and $K_i$ have the same HOMFLY polynomial up to given $z$-degrees, for all braided patterns $P$ with winding number less than or equal to $w$.
The genus non-increasing totally positive unknotting number is the minimum number of crossing changes that transform a knot into the unknot, such that all the crossing changes are positive-to-negative crossing changes that do not increase the genus. We show that the genus non-increasing totally positive unknotting number can be arbitrary large for genus one knots.
A link in $S^{3}$ is a fully positive braid link if it is the closure of a positive braid that contains at least one full-twist. We show that a fully positive braid link is a satellite link if and only if it is the satellite of a fully positive braid link $C$ such that the pattern is a positive braid that contains sufficiently many full twists, where the number of necessary full twists only depends on $C$. As an application, we give a characterization of the unknot by the property that certain braided satellite is a (fully) positive braid knot.
The author would like to report an error of [T. Ito, Proc. Amer. Math. Soc. 152 (2024), pp. 4515--4519.] (version 1 of the arXiv:2311.08676) )
We prove that a positive two-bridge knot other than the $(2,k)$ torus knot does not admit chirally cosmetic surgeries, a pair of Dehn surgeries along distinct slopes yielding orientation-reversingly homeomorphic 3-manifolds.
A non-trivial element of a group is a generalized torsion element if some products of its conjugates is the identity. The minimum number of such conjugates is called a generalized torsion order. We provide several restrictions for generalized torsion orders by using $G$-invariant norm and Alexander polynomials.
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 $G(K) = \pi_1(E(K))$ to $\pi_1(K(r))$, 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 $\mathcal{S}_K(\mu)$ for the meridian $\mu$ 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.
We show that a group whose generalized torsion elements are torsion elements (which we call a $TR^{*}$-group) is torsion-by-$R^{*}$ group, an extension of torsion group by a group without generalized torsion elements. We also discuss a generalized torsion group, a group all of whose non-trivial elements are generalized torsion elements.
We show that a special alternating knot with sufficiently large number (more than $63$) of twist regions has no chirally cosmetic surgeries, a pair of Dehn surgeries producing orientation-reversingly homeomorphic $3$-manifolds. In the course of proof, we provide the optimal upper bounds of the primitive finite type invariants of degree 2 and 3 that solve Willerton's conjecture.