arXiv subjects

Mikami Hirasawa

Publications and source records attributed to Mikami Hirasawa.

13 recordsLinked to original sources

arXiv2024

The equivariant genera of marked strongly invertible knots associated with $2$-bridge knots

A marked strongly invertible knot is a triple $(K,h,δ)$ of a knot $K$ in $S^3$, a strong inversion $h$ of $K$, and a subarc $δ\subset \operatorname{Fix}(h)\cong S^1$ bounded by $\operatorname{Fix}(h)\cap K\cong S^0$. An invariant Seifert surface for $(K,h,δ)$ is an $h$-invariant Seifert surface for $K$ that intersects $\operatorname{Fix}(h)$ in the arc $δ$. In this paper, we completely determine the equivariant genus (the minimum of the genera of invariant Seifert surfaces for $(K,h,δ)$) of every marked strongly invertible knot $(K,h,δ)$ with $K$ a $2$-bridge knot.

math.GT↗

arXiv2023

Saddle point braids of braided fibrations and pseudo-fibrations

Let $g_t$ be a loop in the space of monic complex polynomials in one variable of fixed degree $n$. If the roots of $g_t$ are distinct for all $t$, they form a braid $B_1$ on $n$ strands. Likewise, if the critical points of $g_t$ are distinct for all $t$, they form a braid $B_2$ on $n-1$ strands. In this paper we study the relationship between $B_1$ and $B_2$. Composing the polynomials $g_t$ with the argument map defines a pseudo-fibration map on the complement of the closure of $B_1$ in $\mathbb{C}\times S^1$, whose critical points lie on $B_2$. We prove that for $B_1$ a T-homogeneous braid and $B_2$ the trivial braid this map can be taken to be a fibration map. In the case of homogeneous braids we present a visualisation of this fact. Our work implies that for every pair of links $L_1$ and $L_2$ there is a mixed polynomial $f:\mathbb{C}^2\to\mathbb{C}$ in complex variables $u$, $v$ and the complex conjugate $\overline{v}$ such that both $f$ and the derivative $f_u$ have a weakly isolated singularity at the origin with $L_1$ as the link of the singularity of $f$ and $L_2$ as a sublink of the link of the singularity of $f_u$.

math.GT↗

arXiv2022

Invariant Seifert surfaces for strongly invertible knots

We study invariant Seifert surfaces for strongly invertible knots, and prove that the gap between the equivariant genus (the minimum of the genera of invariant Seifert surfaces) of a strongly invertible knot and the (usual) genus of the underlying knot can be arbitrary large. This forms a sharp contrast with Edmonds' theorem that every periodic knot admits an invariant minimal genus Seifert surface. We also prove variants of Edmonds' theorem, which are useful in studying invariant Seifert surfaces for strongly invertible knots.

math.GT↗

arXiv2021

Construction and decomposition of knots as Murasugi sums of Seifert surfaces

A fixed knot $K$ acts via Murasugi sum on the space $\mathcal{S}$ of isotopy classes of knots. This operation endows $\mathcal{S}$ with a directed graph structure denoted by $M\kern-1pt SG(K)$. We show that any given family of knots in $M\kern-1pt SG(K)$ has the structure of a bi-directed complete graph, which is not the case if we restrict the complexity of Murasugi sums. For that purpose, we show that any knot is a Murasugi sum of any two knots, and we give lower and upper bounds for the minimal complexity of Murasugi sum to obtain $K_3$ by $K_1$ and $K_2$. As an application, we show that given any three knots, there is a braid for one knot which splits along a string into braids for the other two knots.

math.GT↗

arXiv2013

Various stabilities of the Alexander polynomials of knots and links

In this paper, we study distribution of the zeros of the Alexander polynomials of knots and links in S^3. We call a knot or link "real stable" (resp. "circular stable") if all the zeros of its Alexander polynomial are real (resp. unit complex). We give a general construction of real stable and circular stable knots and links. We also study pairs of real stable knots and links such that the zeros of the Alexander polynomials are interlaced.

math.GT↗

arXiv2009

Twisted Alexander polynomials of 2-bridge knots associated to metabelian representations

Suppose the knot group G(K) of a knot K has a non-abelian representation ρon A_4 \subset GL(4,Z). We conjecture that the twisted Alexander polynomial of K associated to ρis of the form: Δ_K(t)/(1-t) ϕ(t^3), where Δ_K (t) is the Alexander polynomial of K and ϕ(t^3) is an integer polynomial in t^3. We prove the conjecture for 2-bridge knots K whose group G(K) can be mapped onto a free product Z/2*Z/3. Later, we discuss more general metabelian representations of the knot groups and propose a similar conjecture on the form of the twisted Alexander polynomials.

math.GT↗

arXiv2009

Twisted Alexander polynomials of 2-bridge knots associated to metacyclic representations

Let p be an odd prime and D_p a dihedral group of order 2p. Let ρ: G(K) --> D_p --> GL(p,Z) be a non-abelian representation of the knot group G(K) of a knot K in 3-sphere. Let Δ_{ρ,K} (t) be the twisted Alexander polynomial of K associated to ρ. Let H(p) is the set of 2-bridge knots K, such that G(K) is mapped onto a non-trivial free product Z/2 * Z/p. Then we prove that for any 2-bridge knot K in H(p), Δ_{ρ,K}(t) is of the form Δ_{K}(t)/(1-t) f(t) f(-t) for some integer polynomial f(t), where Δ_K (t) is the Alexander polynomial of K. Further, it is proved that f(t) \equiv {Δ_K (t)/(1+t)}^n (mod p). Later we discuss the twisted Alexander polynomial associated to the general metacyclic representation.

math.GT↗

arXiv2008

Evaluations of the twisted Alexander polynomials of 2-bridge knots at $\pm 1$

Let $H(p)$ be the set of 2-bridge knots $K$ whose group $G$ is mapped onto a non-trivial free product, $Z/2 * Z/p$, $p$ being odd. Then there is an algebraic integer $s_0$ such that for any $K$ in $H(p)$, $G$ has a parabolic representation $ρ$ into $SL(2, Z[s_0]) \subset SL(2,C)$. Let $Δ(t)$ be the twisted Alexander polynomial associated to $ρ$. Then we prove that for any $K$ in $H(p)$, $Δ(1)=-2s_0^{-1}$ and $Δ(-1)=-2s_0^{-1}μ^2$, where $s_0^{-1}, μ\in Z[s_0]$. The number $μ$ can be recursively evaluated.

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arXiv2007

When does a satellite knot fiber?

Necessary and sufficient conditions are given for a satellite knot to be fibered. Any knot $\tilde k$ embeds in an unknotted solid torus $\tilde V$ with arbitrary winding number in such a way that no satellite knot with pattern $(\tilde V, \tilde k)$ is fibered. In particular, there exist nonfibered satellite knots with fibered pattern and companion knots and nonzero winding number.

math.GT↗

arXiv2005

Seifert surfaces in open books, and a new coding algorithm for links

We introduce a new standard form of a Seifert surface $F$. In that standard form, $F$ is obtained by successively plumbing flat annuli to a disk $D$, where the gluing regions are all in $D$. We show that any link has a Seifert surface in the standard form, and thereby present a new way of coding a link. We present an algorithm to read the code directly from a braid presentation.

math.GT↗

arXiv2005

Crosscap numbers of 2-bridge knots

We present a practical algorithm to determine the minimal genus of non-orientable spanning surfaces for 2-bridge knots, called the crosscap numbers. We will exhibit a table of crosscap numbers of 2-bridge knots up to 12crossings (all 362 of them).

math.GT↗

arXiv2003

Constructions of Morse maps for knots and links, and upper bounds on the Morse-Novikov number

The Morse-Novikov number MN(L) of an oriented link L in the 3-sphere is the minimum number of critical points of a Morse map from the complement of L in the 3-sphere to the circle representing the class of a Seifert surface for L (e.g., the Morse-Novikov number of L is zero if and only if L is fibered). We develop various constructions of Morse maps (Milnor maps, Stallings twists, splicing along a link which is a closed braid with respect to a Morse map, Murasugi sums, cutting a Morse map along an arc on a page) and use them to bound Morse-Novikov numbers from above in terms of other knot and link invariants (free genus, crossing number, braid index, wrapping genus and layered wrapping genus).

math.GT↗

arXiv1999

Almost alternating diagrams and fibered links in S^3

Let $L$ be an oriented link with an alternating diagram $D$. It is known that $L$ is a fibered link if and only if the surface $R$ obtained by applying Seifert's algorithm to $D$ is a Hopf plumbing. Here, we call $R$ a Hopf plumbing if $R$ is obtained by successively plumbing finite number of Hopf bands to a disk. In this paper, we discuss its extension so that we show the following theorem. Let $R$ be a Seifert surface obtained by applying Seifert's algorithm to an almost alternating diagrams. Then $R$ is a fiber surface if and only if $R$ is a Hopf plumbing. We also show that the above theorem can not be extended to 2-almost alternating diagrams, that is, we give examples of 2-almost alternating diagrams for knots whose Seifert surface obtained by Seifert's algorithm are fiber surfaces that are not Hopf plumbing. This is shown by using a criterion of Melvin-Morton.

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