## Seminar information archive

Seminar information archive ～05/26｜Today's seminar 05/27 | Future seminars 05/28～

### 2014/06/30

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

Primitive automorphisms of positive entropy of rational and Calabi-Yau threefolds (JAPANESE)

**Keiji Oguiso**(Osaka University)Primitive automorphisms of positive entropy of rational and Calabi-Yau threefolds (JAPANESE)

#### Algebraic Geometry Seminar

15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Invariant subrings of the Cox rings of K3surfaces by automorphism groups (JAPANESE)

**Akiyoshi Sannai**(University of Tokyo)Invariant subrings of the Cox rings of K3surfaces by automorphism groups (JAPANESE)

[ Abstract ]

Cox rings were introduced by D.Cox and are important rings which appeared in algebraic geometry. One of the main topic related with Cox rings is the finite generation of them. In this talk, we consider the Cox rings of K3 surfaces and answer the following question asked by D. Huybrechts; Are the invariant subrings of the Cox rings of K3 surfaces by automorphism groups finitely generated in general?

Cox rings were introduced by D.Cox and are important rings which appeared in algebraic geometry. One of the main topic related with Cox rings is the finite generation of them. In this talk, we consider the Cox rings of K3 surfaces and answer the following question asked by D. Huybrechts; Are the invariant subrings of the Cox rings of K3 surfaces by automorphism groups finitely generated in general?

#### Kavli IPMU Komaba Seminar

16:30-18:00 Room #002 (Graduate School of Math. Sci. Bldg.)

On some quadratic algebras with applications to Topology,

Algebra, Combinatorics, Schubert Calculus and Integrable Systems. (ENGLISH)

**Anatol Kirillov**(RIMS, Kyoto University)On some quadratic algebras with applications to Topology,

Algebra, Combinatorics, Schubert Calculus and Integrable Systems. (ENGLISH)

[ Abstract ]

The main purpose of my talk is to draw attention of the

participants of the seminar to a certain family of quadratic algebras

which has a wide range of applications to the subject mentioned in the

title of my talk.

The main purpose of my talk is to draw attention of the

participants of the seminar to a certain family of quadratic algebras

which has a wide range of applications to the subject mentioned in the

title of my talk.

### 2014/06/28

#### Harmonic Analysis Komaba Seminar

13:30-17:00 Room #128 (Graduate School of Math. Sci. Bldg.)

On the multilinear restriction problem (ENGLISH)

John-Nirenberg lemmas for a doubling measure (ENGLISH)

**Neal Bez**(埼玉大学) 13:30-15:00On the multilinear restriction problem (ENGLISH)

[ Abstract ]

I will discuss the multilinear restriction problem for the Fourier transform. This will include an overview of the pioneering work of Bennett, Carbery and Tao on this problem and the very losely connected multilinear Kakeya problem. I will also discuss some of my own work in this area which is connected to nonlinear Brascamp-Lieb inequalities (joint work with Jonathan Bennett).

I will discuss the multilinear restriction problem for the Fourier transform. This will include an overview of the pioneering work of Bennett, Carbery and Tao on this problem and the very losely connected multilinear Kakeya problem. I will also discuss some of my own work in this area which is connected to nonlinear Brascamp-Lieb inequalities (joint work with Jonathan Bennett).

**Hong Yue**(Georgia College and State University) 15:30-17:00John-Nirenberg lemmas for a doubling measure (ENGLISH)

[ Abstract ]

We study, in the context of doubling metric measure spaces, a class of BMO type functions defined by John and Nirenberg. In particular, we present a new version of the Calderon-Zygmund decomposition in metric spaces and use it to prove the corresponding John-Nirenberg inequality.

We study, in the context of doubling metric measure spaces, a class of BMO type functions defined by John and Nirenberg. In particular, we present a new version of the Calderon-Zygmund decomposition in metric spaces and use it to prove the corresponding John-Nirenberg inequality.

### 2014/06/26

#### Geometry Colloquium

10:00-11:30 Room #122 (Graduate School of Math. Sci. Bldg.)

Entropic curvature-dimension condition and Bochner’s inequality (JAPANESE)

**Kazumasa Kuwada**(Tokyo Institute of Technology)Entropic curvature-dimension condition and Bochner’s inequality (JAPANESE)

[ Abstract ]

As a characterization of "lower Ricci curvature bound and upper dimension bound”, there appear several conditions which make sense even on singular spaces. In this talk we show the equivalence in complete generality between two major conditions: a reduced version of curvature-dimension bounds of Sturm-Lott-Villani via entropy and optimal transport and Bakry–¥'Emery's one via Markov generator or the associated heat semigroup. More precisely, it holds for metric measure spaces where Cheeger's L^2-energy functional is a quadratic form. In particular, we establish the full Bochner inequality, which originally comes from the Bochner-Weitzenb¥"ock formula, on such spaces. This talk is based on a joint work with M. Erbar and K.-T. Sturm (Bonn).

As a characterization of "lower Ricci curvature bound and upper dimension bound”, there appear several conditions which make sense even on singular spaces. In this talk we show the equivalence in complete generality between two major conditions: a reduced version of curvature-dimension bounds of Sturm-Lott-Villani via entropy and optimal transport and Bakry–¥'Emery's one via Markov generator or the associated heat semigroup. More precisely, it holds for metric measure spaces where Cheeger's L^2-energy functional is a quadratic form. In particular, we establish the full Bochner inequality, which originally comes from the Bochner-Weitzenb¥"ock formula, on such spaces. This talk is based on a joint work with M. Erbar and K.-T. Sturm (Bonn).

### 2014/06/25

#### Operator Algebra Seminars

16:45-18:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Supersymmetric C*-dynamical systems (JAPANESE)

**Hajime Moriya**(Shibaura Inst. Technology)Supersymmetric C*-dynamical systems (JAPANESE)

#### Mathematical Biology Seminar

14:50-16:20 Room #128 (Graduate School of Math. Sci. Bldg.)

Mathematical modeling and classification of tumor immunity in cell transfer therapy (JAPANESE)

**Shinnji Nakaoka**(理化学研究所統合生命医科学研究センター)Mathematical modeling and classification of tumor immunity in cell transfer therapy (JAPANESE)

#### Number Theory Seminar

16:40-17:40 Room #056 (Graduate School of Math. Sci. Bldg.)

Periods of some two dimensional reducible p-adic representations and non-de Rham B-pairs (JAPANESE)

**Masahiko Takiguchi**(University of Tokyo)Periods of some two dimensional reducible p-adic representations and non-de Rham B-pairs (JAPANESE)

### 2014/06/24

#### PDE Real Analysis Seminar

10:30-11:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Sudden directional diffusion: counting and watching facets (ENGLISH)

**Piotr Rybka**(University of Warsaw)Sudden directional diffusion: counting and watching facets (ENGLISH)

[ Abstract ]

We study two examples of singular parabolic equations such that the diffusion is so strong that is leads to creation of facets. By facets we mean flat parts of the graphs of solutions with singular slopes. In one of the equations we study there are two singular slopes. The other equation has just one singular slope and the isotropic diffusion term. For both problems we watch and count facet.

For the system with two singular slopes a natural question arises if any solution may have an infinite number of oscillations. We also show that the solutions we constructed are viscosity solutions. This in turn gives estimates on the extinction time based on the comparison principle.

We study two examples of singular parabolic equations such that the diffusion is so strong that is leads to creation of facets. By facets we mean flat parts of the graphs of solutions with singular slopes. In one of the equations we study there are two singular slopes. The other equation has just one singular slope and the isotropic diffusion term. For both problems we watch and count facet.

For the system with two singular slopes a natural question arises if any solution may have an infinite number of oscillations. We also show that the solutions we constructed are viscosity solutions. This in turn gives estimates on the extinction time based on the comparison principle.

#### Tuesday Seminar on Topology

17:10-18:10 Room #056 (Graduate School of Math. Sci. Bldg.)

On third homologies of quandles and of groups via Inoue-Kabaya map (JAPANESE)

**Takefumi Nosaka**(Faculty of Mathematics, Kyushu University)On third homologies of quandles and of groups via Inoue-Kabaya map (JAPANESE)

[ Abstract ]

本講演では, 群とその群同型の組から定まるカンドルを扱い, 以下の結果を紹介

する. まず, その際Inoue-Kabaya鎖写像が, カンドルホモロジーから群ホモロ

ジーへの写像とし, 定式化される事を見る. 例えば, 有限体上のAlexander

quandleに対し, 望月3-コサイクル全ては, 当写像を通じ, 或る群コホモロジー

から導出され, 殆どがトリプルマッセイ積で解釈できる事をみる. 加えてカンド

ルの普遍中心拡大に対し, Inoue-Kabaya鎖写像が3次において(或る捩れ部分を

除き)同型となる. なお講演内容は当週にある集中講義の聴講を仮定しない.

本講演では, 群とその群同型の組から定まるカンドルを扱い, 以下の結果を紹介

する. まず, その際Inoue-Kabaya鎖写像が, カンドルホモロジーから群ホモロ

ジーへの写像とし, 定式化される事を見る. 例えば, 有限体上のAlexander

quandleに対し, 望月3-コサイクル全ては, 当写像を通じ, 或る群コホモロジー

から導出され, 殆どがトリプルマッセイ積で解釈できる事をみる. 加えてカンド

ルの普遍中心拡大に対し, Inoue-Kabaya鎖写像が3次において(或る捩れ部分を

除き)同型となる. なお講演内容は当週にある集中講義の聴講を仮定しない.

#### Classical Analysis

16:00-17:30 Room #122 (Graduate School of Math. Sci. Bldg.)

Irreducibility of the discrete Painlev\\'e equation of type $D_7$ (JAPANESE)

**Nishioka Seiji**(Yamagata University)Irreducibility of the discrete Painlev\\'e equation of type $D_7$ (JAPANESE)

### 2014/06/23

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

A remark to the division algorithm in the proof of Oka's First Coherence Theorem (JAPANESE)

**Junjiro Noguchi**(University of Tokyo)A remark to the division algorithm in the proof of Oka's First Coherence Theorem (JAPANESE)

[ Abstract ]

The problem is the local finite generation of a relation sheaf $R(f_1, \ldots, f_q)$ in $\mathcal{O}_n=\mathcal{O}_{C^n}$. After $f_j$ reduced to Weierstrass' polynomials in $z_n$, it is the key to apply the induction in $n$ to show that elements of $R(f_1, \ldots, q)$ are expressed by $z_n$-polynomial-like elements of degree at most $p=\max_j\deg f_j$ over $\mathcal{O}_n$. In that proof one is used to use a divison by $f_j$ of $\deg f_j=p$ (Oka '48, Cartan '50, Hörmander, Demailly, . . .). In this talk we shall confirm that the division abve works by making use of $f_k$ of the minimum degree $\min_j \deg f_j$. This proof is natrually compatible with the simple case when some $f_j$ is a unit, and gives some improvement in the degree estimate of generators.

The problem is the local finite generation of a relation sheaf $R(f_1, \ldots, f_q)$ in $\mathcal{O}_n=\mathcal{O}_{C^n}$. After $f_j$ reduced to Weierstrass' polynomials in $z_n$, it is the key to apply the induction in $n$ to show that elements of $R(f_1, \ldots, q)$ are expressed by $z_n$-polynomial-like elements of degree at most $p=\max_j\deg f_j$ over $\mathcal{O}_n$. In that proof one is used to use a divison by $f_j$ of $\deg f_j=p$ (Oka '48, Cartan '50, Hörmander, Demailly, . . .). In this talk we shall confirm that the division abve works by making use of $f_k$ of the minimum degree $\min_j \deg f_j$. This proof is natrually compatible with the simple case when some $f_j$ is a unit, and gives some improvement in the degree estimate of generators.

### 2014/06/19

#### Geometry Colloquium

10:00-11:30 Room #122 (Graduate School of Math. Sci. Bldg.)

Antipodal structure of the intersection of real forms and its applications (JAPANESE)

**Takashi Sakai**(Tokyo Metropolitan University)Antipodal structure of the intersection of real forms and its applications (JAPANESE)

[ Abstract ]

A subset A of a Riemannian symmetric space is called an antipodal set if the geodesic symmetry s_x fixes all points of A for each x in A. This notion was first introduced by Chen and Nagano. Tanaka and Tasaki proved that the intersection of two real forms L_1 and L_2 in a Hermitian symmetric space of compact type is an antipodal set of L_1 and L_2. As an application, we calculate the Lagrangian Floer homology of a pair of real forms in a monotone Hermitian symmetric space. Then we obtain a generalization of the Arnold-Givental inequality. We expect to generalize the above results to the case of complex flag manifolds. In fact, using the k-symmetric structure, we can describe an antipodal set of a complex flag manifold. Moreover we can observe the antipodal structure of the intersection of certain real forms in a complex flag manifold.

This talk is based on a joint work with Hiroshi Iriyeh and Hiroyuki Tasaki.

A subset A of a Riemannian symmetric space is called an antipodal set if the geodesic symmetry s_x fixes all points of A for each x in A. This notion was first introduced by Chen and Nagano. Tanaka and Tasaki proved that the intersection of two real forms L_1 and L_2 in a Hermitian symmetric space of compact type is an antipodal set of L_1 and L_2. As an application, we calculate the Lagrangian Floer homology of a pair of real forms in a monotone Hermitian symmetric space. Then we obtain a generalization of the Arnold-Givental inequality. We expect to generalize the above results to the case of complex flag manifolds. In fact, using the k-symmetric structure, we can describe an antipodal set of a complex flag manifold. Moreover we can observe the antipodal structure of the intersection of certain real forms in a complex flag manifold.

This talk is based on a joint work with Hiroshi Iriyeh and Hiroyuki Tasaki.

### 2014/06/18

#### Operator Algebra Seminars

16:30-18:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Toward the classification of irreducible unitary spherical representations of the Drinfeld double of $SU_q(3)$ (ENGLISH)

**Yuki Arano**(Univ. Tokyo)Toward the classification of irreducible unitary spherical representations of the Drinfeld double of $SU_q(3)$ (ENGLISH)

### 2014/06/17

#### Tuesday Seminar on Topology

16:30-18:00 Room #002 (Graduate School of Math. Sci. Bldg.)

Bounded Euler number of actions of 2-orbifold groups on the circle (JAPANESE)

**Yoshifumi Matsuda**(Aoyama Gakuin University)Bounded Euler number of actions of 2-orbifold groups on the circle (JAPANESE)

[ Abstract ]

Burger, Iozzi and Wienhard defined the bounded Euler number for a

continuous action of the fundamental group of a connected oriented

surface of finite type possibly with punctures on the circle. A Milnor-Wood

type inequality involving the bounded Euler number holds and its maximality

characterizes Fuchsian actions up to semiconjugacy. The definition of the

bounded Euler number can be extended to actions of 2-orbifold groups by

considering coverings. A Milnor-Wood type inequality and the characterization

of Fuchsian actions also hold in this case. In this talk, we describe when lifts

of Fuchsian actions of certain 2-orbifold groups, such as the modular group,

are characterized by its bounded Euler number.

Burger, Iozzi and Wienhard defined the bounded Euler number for a

continuous action of the fundamental group of a connected oriented

surface of finite type possibly with punctures on the circle. A Milnor-Wood

type inequality involving the bounded Euler number holds and its maximality

characterizes Fuchsian actions up to semiconjugacy. The definition of the

bounded Euler number can be extended to actions of 2-orbifold groups by

considering coverings. A Milnor-Wood type inequality and the characterization

of Fuchsian actions also hold in this case. In this talk, we describe when lifts

of Fuchsian actions of certain 2-orbifold groups, such as the modular group,

are characterized by its bounded Euler number.

#### Number Theory Seminar

17:30-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Vinberg's monoid and automorphic L-functions (ENGLISH)

**Bao Châu Ngô**(University of Chicago, VIASM)Vinberg's monoid and automorphic L-functions (ENGLISH)

[ Abstract ]

We will explain a generalisation of the construction of the local factors of Godement-Jacquet's L-functions, based on Vinberg's monoid.

We will explain a generalisation of the construction of the local factors of Godement-Jacquet's L-functions, based on Vinberg's monoid.

#### Lie Groups and Representation Theory

16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)

SINGULAR EQUIVARIANT ASYMPTOTICS AND THE MOMENTUM MAP. RESIDUE FORMULAE IN EQUIVARIANT COHOMOLOGY (ENGLISH)

**Pablo Ramacher**(Marburg University)SINGULAR EQUIVARIANT ASYMPTOTICS AND THE MOMENTUM MAP. RESIDUE FORMULAE IN EQUIVARIANT COHOMOLOGY (ENGLISH)

[ Abstract ]

Let M be a smooth manifold and G a compact connected Lie group acting on M by isometries. In this talk, we study the equivariant cohomology of the cotangent bundle of M, and relate it to the cohomology of the Marsden-Weinstein reduced space via certain residue formulae. In case of compact symplectic manifolds with a Hamiltonian G-action, similar residue formulae were derived by Jeffrey, Kirwan et al..

Let M be a smooth manifold and G a compact connected Lie group acting on M by isometries. In this talk, we study the equivariant cohomology of the cotangent bundle of M, and relate it to the cohomology of the Marsden-Weinstein reduced space via certain residue formulae. In case of compact symplectic manifolds with a Hamiltonian G-action, similar residue formulae were derived by Jeffrey, Kirwan et al..

### 2014/06/16

#### Seminar on Geometric Complex Analysis

10:30-12:00 Room #126 (Graduate School of Math. Sci. Bldg.)

New examples of weighted Bergman kernels on a certain non-homogeneous Siegel domain (JAPANESE)

**Hideyuki Ishi**(Nagoya University)New examples of weighted Bergman kernels on a certain non-homogeneous Siegel domain (JAPANESE)

#### Kavli IPMU Komaba Seminar

16:30-18:00 Room #002 (Graduate School of Math. Sci. Bldg.)

Universal formulae for Lie groups and Chern-Simons theory (ENGLISH)

**A.P. Veselov**(Loughborough, UK and Tokyo)Universal formulae for Lie groups and Chern-Simons theory (ENGLISH)

[ Abstract ]

In 1990s Vogel introduced an interesting parametrization of simple

Lie algebras by 3 parameters defined up to a common multiple and

permutations. Numerical characteristic is called universal if it can be

expressed in terms of Vogel's parameters (example - the dimension of Lie

algebra). I will discuss some universal formulae for Lie groups

and Chern-Simons theory on 3D sphere.

The talk is based on joint work with R.L. Mkrtchyan and A.N. Sergeev.

In 1990s Vogel introduced an interesting parametrization of simple

Lie algebras by 3 parameters defined up to a common multiple and

permutations. Numerical characteristic is called universal if it can be

expressed in terms of Vogel's parameters (example - the dimension of Lie

algebra). I will discuss some universal formulae for Lie groups

and Chern-Simons theory on 3D sphere.

The talk is based on joint work with R.L. Mkrtchyan and A.N. Sergeev.

### 2014/06/11

#### Operator Algebra Seminars

16:30-18:00 Room #118 (Graduate School of Math. Sci. Bldg.)

Finiteness of K-area and the dual of the Baum-Connes conjecture (ENGLISH)

**Yosuke Kubota**(Univ. Tokyo)Finiteness of K-area and the dual of the Baum-Connes conjecture (ENGLISH)

#### Mathematical Biology Seminar

14:50-16:20 Room #128 (Graduate School of Math. Sci. Bldg.)

)

Path Integral Formulation and Variational Structure in Multitype Population Dynamics

(JAPANESE)

**Tetsuya Kobayashi**(Center for Research on Integrated Biomedical Systems, Institute of Industrial Science, the University of Tokyo)

Path Integral Formulation and Variational Structure in Multitype Population Dynamics

(JAPANESE)

### 2014/06/10

#### Lectures

14:40-16:10 Room #056 (Graduate School of Math. Sci. Bldg.)

Bipartite knots (ENGLISH)

**Sergei Duzhin**(Steklov Institute of Mathematics)Bipartite knots (ENGLISH)

[ Abstract ]

We give a solution to a part of Problem 1.60 in Kirby's list of open

problems in topology thus proving a conjecture raised in 1987 by

J.Przytycki. A knot is said to be bipartite if it has a "matched" diagram,

that is, a plane diagram that has an even number of crossings which can be

split into pairs that look like a simple braid on two strands with two

crossings. The conjecture was that there exist knots that do not have such

diagrams. I will prove this fact using higher Alexander ideals.

This talk is based on a joint work with my student M.Shkolnikov

We give a solution to a part of Problem 1.60 in Kirby's list of open

problems in topology thus proving a conjecture raised in 1987 by

J.Przytycki. A knot is said to be bipartite if it has a "matched" diagram,

that is, a plane diagram that has an even number of crossings which can be

split into pairs that look like a simple braid on two strands with two

crossings. The conjecture was that there exist knots that do not have such

diagrams. I will prove this fact using higher Alexander ideals.

This talk is based on a joint work with my student M.Shkolnikov

#### Seminar on Mathematics for various disciplines

10:30-11:30 Room #056 (Graduate School of Math. Sci. Bldg.)

The self-similar collapse solution of a point vortex system and complex time singularities (JAPANESE)

**Yoshifumi Kimura**(Graduate School of Mathematics, Nagoya University)The self-similar collapse solution of a point vortex system and complex time singularities (JAPANESE)

[ Abstract ]

A system of N point vortices is a Hamiltonian dynamical system with N degrees of freedom,and it is known that under certain parameter and initial conditions, there are self-similar collapse solutions for which N vortices collide at a point while rotating without changing the initial shape of configuration. In this talk, I will introduce such collision solutions and discuss some properties of complex time singularities in relation with those solutions.

A system of N point vortices is a Hamiltonian dynamical system with N degrees of freedom,and it is known that under certain parameter and initial conditions, there are self-similar collapse solutions for which N vortices collide at a point while rotating without changing the initial shape of configuration. In this talk, I will introduce such collision solutions and discuss some properties of complex time singularities in relation with those solutions.

#### Tuesday Seminar of Analysis

16:30-18:00 Room #128 (Graduate School of Math. Sci. Bldg.)

On estimates for the Stokes flow in a space of bounded functions (JAPANESE)

**Ken Abe**(Nagoya University)On estimates for the Stokes flow in a space of bounded functions (JAPANESE)

[ Abstract ]

The Stokes equations are well understood on $L^p$ space for various kinds of domains such as bounded or exterior domains, and fundamental to study the nonlinear Navier-Stokes equations. The situation is different for the case $p=\\infty$ since in this case the Helmholtz projection does not act as a bounded operator anymore. In this talk, we show some a priori estimate for a composition operator of the Stokes semigroup and the Helmholtz projection on a space of bounded functions.

The Stokes equations are well understood on $L^p$ space for various kinds of domains such as bounded or exterior domains, and fundamental to study the nonlinear Navier-Stokes equations. The situation is different for the case $p=\\infty$ since in this case the Helmholtz projection does not act as a bounded operator anymore. In this talk, we show some a priori estimate for a composition operator of the Stokes semigroup and the Helmholtz projection on a space of bounded functions.

#### Tuesday Seminar on Topology

16:30-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

On relation between the Milnor's $¥mu$-invariant and HOMFLYPT

polynomial (JAPANESE)

**Yuka Kotorii**(The University of Tokyo)On relation between the Milnor's $¥mu$-invariant and HOMFLYPT

polynomial (JAPANESE)

[ Abstract ]

Milnor introduced a family of invariants for ordered oriented link,

called $¥bar{¥mu}$-invariants. Polyak showed a relation between the $¥

bar{¥mu}$-invariant of length 3 sequence and Conway polynomial.

Moreover, Habegger-Lin showed that Milnor's invariants are invariants of

string link, called $¥mu$-invariants. We show that any $¥mu$-invariant

of length $¥leq k$ can be represented as a combination of HOMFLYPT

polynomials if all $¥mu$-invariant of length $¥leq k-2$ vanish.

This result is an extension of Polyak's result.

Milnor introduced a family of invariants for ordered oriented link,

called $¥bar{¥mu}$-invariants. Polyak showed a relation between the $¥

bar{¥mu}$-invariant of length 3 sequence and Conway polynomial.

Moreover, Habegger-Lin showed that Milnor's invariants are invariants of

string link, called $¥mu$-invariants. We show that any $¥mu$-invariant

of length $¥leq k$ can be represented as a combination of HOMFLYPT

polynomials if all $¥mu$-invariant of length $¥leq k-2$ vanish.

This result is an extension of Polyak's result.

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