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Masaaki Suzuki

Publications and source records attributed to Masaaki Suzuki.

33 records · Page 2Linked to original sources
arXiv2016

Epimorphisms between $2$-bridge knot groups and their crossing numbers

Suppose that there exists an epimorphism from the knot group of a $2$-bridge knot $K$ onto that of another knot $K'$. In this paper, we study the relationship between their crossing numbers $c(K)$ and $c(K')$. Especially it is shown that $c(K)$ is greater than or equal to $3 c(K')$ and we estimate how many knot groups a $2$-bridge knot group maps onto. Moreover, we formulate the generating function which determines the number of $2$-bridge knot groups admitting epimorphisms onto the knot group of a given $2$-bridge knot.

math.GT↗
arXiv2016

On minimality of two-bridge knots

A knot is called minimal if its knot group admits epimorphisms onto the knot groups of only the trivial knot and itself. In this paper, we determine which two-bridge knot $\mathfrak{b}(p,q)$ is minimal where $q \leq 6$ or $p \leq 100$.

math.GT↗
arXiv2016

An abelian quotient of the symplectic derivation Lie algebra of the free Lie algebra

We construct an abelian quotient of the symplectic derivation Lie algebra $\mathfrak{h}_{g,1}$ of the free Lie algebra generated by the fundamental representation of $\mathrm{Sp}(2g,\mathbb{Q})$. More specifically, we show that the weight $12$ part of the abelianization of $\mathfrak{h}_{g,1}$ is $1$-dimensional for $g \ge 8$. The computation is done with the aid of computers.

math.AT↗
arXiv2016

Secondary characteristic classes for subgroups of automorphism groups of free groups

By analyzing how the Borel regulator classes vanish on various groups related to $\mathrm{GL}(n,\mathrm{Z})$, we define three series of secondary characteristic classes for subgroups of automorphism groups of free groups. The first case is the $\mathrm{IA}$-automorphism groups and we show that our classes coincide with higher $\mathrm{FR}$ torsions due to Igusa. The second case is the mapping class groups and our classes also turn out to be his higher torsions which are non-zero multiples of the Mumford-Morita-Miller classes of even indices. Our construction gives new group cocycles for these still mysterious classes. The third case is the outer automorphism groups of free groups of specific ranks. Here we give a conjectural geometric meaning to a series of unstable homology classes called the Morita classes. We expect that certain unstable secondary classes would detect them.

math.AT↗
arXiv2015

Structure of symplectic invariant Lie subalgebras of symplectic derivation Lie algebras

We study the structure of the symplectic invariant part $\mathfrak{h}_{g,1}^{\mathrm{Sp}}$ of the Lie algebra $\mathfrak{h}_{g,1}$ consisting of symplectic derivations of the free Lie algebra generated by the rational homology group of a closed oriented surface $Σ_{g}$ of genus $g$. First we describe the orthogonal direct sum decomposition of this space which is induced by the canonical metric on it and compute it explicitly up to degree $20$. In this framework, we give a general constraint which is imposed on the $\mathrm{Sp}$-invariant component of the bracket of two elements in $\mathfrak{h}_{g,1}$. Second we clarify the relations among $\mathfrak{h}_{g,1}$ and the other two related Lie algebras $\mathfrak{h}_{g,*}$ and $\mathfrak{h}_{g}$ which correspond to the cases of a closed surface $Σ_g$ with and without base point $*\inΣ_g$. In particular, based on a theorem of Labute, we formulate a method of determining these differences and describe them explicitly up to degree $20$. Third, by giving a general method of constructing elements of $\mathfrak{h}_{g,1}^{\mathrm{Sp}}$, we reveal a considerable difference between the two submodules of it, one is the $\mathrm{Sp}$-invariant part of a certain ideal $\mathfrak{j}_{g,1}$ and the other is that of the Johnson image. Finally we combine these results to determine the structure of $\mathfrak{h}_{g,1}$ completely up to degree $6$ including the unstable cases where the genus $1$ case has an independent meaning. In particular, we see a glimpse of the Galois obstructions explicitly from our point of view.

math.AT↗
arXiv2015

Computations in formal symplectic geometry and characteristic classes of moduli spaces

We make explicit computations in the formal symplectic geometry of Kontsevich and determine the Euler characteristics of the three cases, namely commutative, Lie and associative ones, up to certain weights.From these, we obtain some non-triviality results in each case. In particular, we determine the integral Euler characteristics of the outer automorphism groups Out F_n of free groups for all n <= 10 and prove the existence of plenty of rational cohomology classes of odd degrees. We also clarify the relationship of the commutative graph homology with finite type invariants of homology 3-spheres as well as the leaf cohomology classes for transversely symplectic foliations. Furthermore we prove the existence of several new non-trivalent graph homology classes of odd degrees. Based on these computations, we propose a few conjectures and problems on the graph homology and the characteristic classes of the moduli spaces of graphs as well as curves.

math.AT↗
arXiv2015

Non-meridional epimorphisms of knot groups

In the literature of the study of knot group epimorphisms, the existence of an epimorphism between two given knot groups is mostly (if not always) shown by giving an epimorphism which preserves meridians. A natural question arises: is there an epimorphism preserving meridians whenever a knot group is a homomorphic image of another? We answer in the negative by presenting infinitely many pairs of prime knot groups (G,G') such that G' is a homomorphic image of G but no epimorphism of G onto G' preserves meridians.

math.GT↗
arXiv2014

Some minimal elements for a partial order of prime knots

A partial order on the set of prime knots can be defined by the existence of an epimorphism between knot groups. We prove that all the prime knots with up to $6$ crossings are minimal. We also show that each fibered knot with the irreducible Alexander polynomial is minimal.

math.GT↗
arXiv2013

Abelianizations of derivation Lie algebras of the free associative algebra and the free Lie algebra

We determine the abelianizations of the following three kinds of graded Lie algebras in certain stable ranges: derivations of the free associative algebra, derivations of the free Lie algebra and symplectic derivations of the free associative algebra. In each case, we consider both the whole derivation Lie algebra and its ideal consisting of derivations with positive degrees. As an application of the last case, and by making use of a theorem of Kontsevich, we obtain a new proof of the vanishing theorem of Harer concerning the top rational cohomology group of the mapping class group with respect to its virtual cohomological dimension.

math.AT↗
arXiv2012

An integral region choice problem on knot projection

In this paper we propose {\it a region choice problem} for a knot projection. This problem is an integral extension of Shimizu's 'region crossing change unknotting operation.' We show that there exists a solution of the region choice problem for all knot projections.

math.GT↗
arXiv2009

A partial order on the set of prime knots with up to 11 crossings

Let $K$ be a prime knot in $S^3$ and $G(K)=π_1(S^3-K)$ the knot group. We write $K_1 \geq K_2$ if there exists a surjective homomorphism from $G(K_1)$ onto $G(K_2)$. In this paper, we determine this partial order on the set of prime knots with up to 11 crossings. There exist such 801 prime knots and then $640,800$ should be considered. The existence of a surjective homomorphism can be proved by constructing it explicitly. On the other hand, the non-existence of a surjective homomorphism can be proved by the Alexander polynomial and the twisted Alexander polynomial. This work is an extension of the result of \cite{KS1}.

math.GT↗
arXiv2006

Spectropolarimetric Study on Circumstellar Structure of Microquasar LS I +61deg 303

We present optical linear spectropolarimetry of the microquasar LS I +61$^{\circ}$ 303. The continuum emission is mildly polarized (up to 1.3 %) and shows almost no temporal change. We find a distinct change of polarization across the H$α$ emission line, indicating the existence of polarization component intrinsic to the microquasar. We estimate the interstellar polarization (ISP) component from polarization of the H$α$ line and derive the intrinsic polarization component. The wavelength dependence of the intrinsic component is well explained by Thomson scattering in equatorial disk of the Be-type mass donor. The position angle (PA) of the intrinsic polarization $\sim 25^{\circ}$ represents the rotational axis of the Be disk. This PA is nearly perpendicular to the PA of the radio jet found during quiescent phases. Assuming an orthogonal disk-jet geometry around the compact star, the rotational axis of the accretion disk is almost perpendicular to that of the Be disk. Moreover, according to the orbital parameters of the microquasar, the compact star is likely to get across the Be disk around their periastron passage. We discuss the peculiar circumstellar structure of this microquasar inferred from our observation and possible connection with its high-energy activities.

astro-ph↗