Citation 数理解析研究所講究録 (2010), 1716:

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1716: 1-5 Issue Date 2010-10 URL http://hdlhandlenet/2433/170319

1 Title AN EXTENSION OF BURAU REPRESENTATIO BRAID GROUPS (Intelligence of Low-d Author(s) MATSUDA, HIROSHI Citation 数理解析研究所講究録 (2010), 1716: 1-5 Issue Date URL Right Type Departmental Bulletin Paper Textversion publisher Kyoto University

2 of AN EXTENSION OF BURAU REPRESENTATION OF THE BRAID GROUPS HIROSHI MATSUDA Artin [3] introduced the braid group $B_{n}=\{\begin{array}{llll} \sigma_{i}\sigma_{j}=\sigma_{j}\sigma_{i}if i-j >1\sigma_{1} \cdots \sigma_{n-1} \sigma_{i}\sigma_{i+1}\sigma_{i}=\sigma_{i+1}\sigma_{i}\sigma_{i+1}\end{array}\}$ Alexander [1] found a connection between the braid group and links in $B_{n}$ $S^{3}$ Markov [10] introduced (three) Markov moves on closed braids, and announced Markov Theorem (Weinberg [13] showed that one of three Markov moves was unneccesary) This theorem says that studying closed braids modulo Markov moves, Markov equivalence classes, is equivalent to studying ambient isotopy classes of links in Birman [6] gave a first complete proof of Markov Theorem $S^{3}$ Around the same time as the announcement of Markov theorem, Burau investigated a connection between a representation of and Alexander polynomial of $B_{n}$ links We introduce Burau representation of in \S 1 $B_{n}$ 1 BURAU REPRESENTATION Burau [7] defined a representation, Burau representation, of the braid group $B_{n}$ $\varphi_{n}(\sigma_{i})$ The image $\sigma_{i}$ of a generator the matrix $\varphi_{n}:b_{n}arrow M(n;\mathbb{Z}[t, t^{-1}])$ $B_{n}$ is represented by $(\begin{array}{llll}i_{i-1} O O OO 1-t t OO 1 0 OO O O I_{n-(i+1)}\end{array})$ This representation $\varphi_{n}$ $\varphi_{n} $ : is reducible, and is reduced to an irreducible representation $B_{n}arrow M(n-1;\mathbb{Z}[t, t^{-1}])$ The images of generators $\sigma_{1},$ $n-2)$ of by $B_{n}$ $\varphi_{n} $ are represented by the following matrices: $\sigma_{n-1}$ and $\sigma_{i}(2\leq i\leq$

3 denote 2 $\varphi_{n} (\sigma_{1})=(\begin{array}{lll}-t 1 O0 1 OO O I_{n-3}\end{array})$ $\varphi_{n} (\sigma_{n-1})=(\begin{array}{lll}i_{i-3} O OO 1 0O t -t\end{array})$ $\varphi_{n}^{f}(\sigma_{i})=(i_{i-2}oooo$ $OO01t$ $-too00$ $OO011$ $I_{n-(i+2)}OOOO)$ $\varphi_{n}$ Remark 1 The representation is faithful when $n\leq 3[9]$, and $n\geq 5[12],$ $[8]$, [5] It is not known whether is faithful $\varphi_{4}:b_{4}arrow M(4;\mathbb{Z}[t, t^{-1}])$ Burau obtained a knot invariant, Alexander polynomial, by measuring how far $\varphi_{n}^{f}(\beta)$ 1 departs from being an eigenvalue of Theorem 2 [7] Let denote a word in Let $B_{n}$ $K$ denote a link in that $S^{3}$ is a closed n-bmid corresponding to Then $\frac{\det(\varphi_{n} (\beta)-i_{n-1})}{\det(\varphi_{n}^{f}(\sigma_{1}\sigma_{2}\cdots\sigma_{n-1})-i_{n-1})}$ is equal to Alexander polynomial of $K,$ $\triangle_{k}(t)$, up to multiplications by $t$ 2 EXTENSION OF BURAU REPRESENTATION In this section, we use matrices instead of in Burau representation, $2\cross 2$ $\mathbb{z}[t, t^{-1}]$ that is, we study a mapping $\psi_{n}:b_{n}arrow M$ ( $n;2\cross 2$ matrices) Let $\Lambda$ a set of elementary functions with variables $a,$ $b,$ $c,$ $d,$ $e,$ $f,$ $g,$ $h$, $p,$ $q,$ $r,$ $s,$ $t,$ $u,$ $v,$ $w$ Let $K=(\begin{array}{ll}a bc d\end{array}),$ $L=(\begin{array}{ll}e fg h\end{array}),$ $M=(\begin{array}{ll}p qr s\end{array}),$ $N=(\begin{array}{ll}t uv w\end{array})$

4 3 denote elements in $GL(2;\Lambda)$ We assume that the 4 $\cross 4$ matrix $(\begin{array}{ll}k LM N\end{array})$ is invertible in $M(4;\Lambda)$ We define a mapping $\psi_{n}:b_{n}arrow M(n;GL(2;\Lambda))$ by $\psi_{n}(\sigma_{i})=(\begin{array}{llll}i_{2(i-l)} O O OO K L OO M N OO O O I_{2(n-(i+1))}\end{array})$ This mapping $\psi_{n}$ is a homomorphism when the following conditions are satisfied; $M=K^{-1}L^{-1}K(I_{2}-K)$, $N=I_{2}-K^{-1}L^{-1}KL$, $h= \frac{1}{acf+b(bc+d-ad)g}\{cf(a(e-de)+c(be+f))$ $+(bde+(a-d)(-a-bc-d+ad)f)g-b^{2}g^{2}\}$, and $e= \frac{1}{2bc(-bc+ad)}\{a^{2}cf+2bc^{2}f-acdf+2b^{2}$ cg ab2 $cg-abdg+a^{2}bdg+b^{2}cdg$ $+bd^{2}$g abd2$g+(acf+b(bc+d-ad)g)\sqrt{4bc+(a-d)^{2}}\}$ Therefore we obtain a representation of $B_{n},$ $\psi_{n}:b_{n}arrow M(n;GL(2;\Lambda))$ This representation $\psi_{n}$ $\psi_{n} $ : is reducible, and is reduced to a representation $B_{n}arrow M(n-1;GL(2;\Lambda))$ Remark 3 The representation $\psi_{n}$ See [4], for example has something to do with a biquandle In a similar method as Burau obtained Alexander polynomial from Burau representation, we obtain a knot invariant from the representation $\psi_{n} $ Theorem 4 [11] Let denote a word in $B_{n}$ Let $K$ denote a link in $S^{3}$ that is a closed n-braid corresponding to $\frac{\det(\psi_{n} (\beta)-i_{2(n-1)})}{\det(\psi_{n}(\sigma_{1}\sigma_{2}\cdots\sigma_{n-1})-i_{2(n-1)})}$ Then is a knot invariant, up to multiplications by $(a-1)(d-1)-bc$

5 $\lambda,$ 4 3 EXAMPLE AND PROBLEM In this section, we denote $\Delta_{2}(\beta)=\frac{\det(\psi_{n} (\beta)-i_{2(n-1)})}{\det(\psi_{n}^{f}(\sigma_{1}\sigma_{2}\cdots\sigma_{n-1})-i_{2(n-1)})}$, and $D=(a-1)(d-1)-bc,$ $T=(a-1)+(d-1)$ Example 5 We calculate $\Delta_{2}(\beta)$ for some numbers of (1) Suppose Then $\beta=\sigma_{1}\in B_{2}$ $K$ is a trivial knot, and $\Delta_{2}(\sigma_{1})=1$ (2) Suppose $\beta=1\in B_{2}$ Then $K$ is a 2-component trivial link, and $\triangle_{2}(1)=0$ (3) Suppose Then $\beta=\sigma_{1}^{2}\in B_{2}$ $K$ is a positive Hopf link, and $\Delta_{2}(\sigma_{1}^{2})=D+T+1$ (4) Suppose Then $\beta=\sigma_{1}^{-2}\in B_{2}$ $K$ is a negative Hopf link, and $\triangle_{2}(\sigma_{1}^{-2})=\frac{1}{d^{2}}\triangle_{2}(\sigma_{1}^{2})$ This is equal to $\triangle_{2}(\sigma_{1}^{2})$, up to multiplications by $D^{2}$ (5) Suppose $\beta=\sigma_{1}^{3}\in B_{2}$ Then $K$ is a right-handed trefoil knot, and $\triangle_{2}(\sigma_{1}^{3})=d^{2}+dt+t^{2}-d+t+1$ (6) Suppose $\beta=\sigma_{1}^{-3}\in B_{2}$ Then $K$ is a left-handed trefoil knot, and $\Delta_{2}(\sigma_{1}^{-3})=\frac{1}{D^{3}}\triangle_{2}(\sigma_{1}^{3})$ This is equal to $\Delta_{2}(\sigma_{1}^{3})$, up to multiplications by $D^{3}$ Observing calculations in Example 5, we pose the following conjecture $\Delta_{2}(\beta)$ Conjecture 6 The knot invariant is $D_{\rangle}$ multiplications by for every $\beta\in B_{n}$ an element in $\mathbb{z}[d, D^{-1}, T, T^{-1}]$, up to In order to extend our extension of Burau representation, we pose the following problem Problem 7 Choose your favorite algebra $\Omega$ $a$ quandle algebm or $GL(n;\Lambda)$ for some algebm $\Lambda$ with unit 1 (For example, $\Omega$ $\kappa,$ ) Let $\mu,$ $\nu$ might be denote elements in $\Omega$ such that $(\begin{array}{ll}\kappa \lambda\mu \nu\end{array})$ is invertible in $M(2;\Omega)$ We define a mapping $\phi_{n}:b_{n}arrow M(n;\Omega)$ by $\phi_{n}(\sigma_{i})=(\begin{array}{llll}i_{i-1} O O OO \kappa \lambda OO \mu \nu OO O O I_{n-(i+1)}\end{array})$ (1) Select $\kappa,$ $\lambda,$ $\mu,$ (We refer to [2] for one answer) $\nu\in\omega$ so that $\phi_{n}:b_{n}arrow M(n;\Omega)$ is a homomorphism

6 5 (2) Let denote a word in Let $B_{n}$ $K$ denote a link in that is a closed n-braid $S^{3}$ $cor\gamma esponding$ to Construct an invariant of Markov equivalence classes of that is, an invariant of $K$ by evaluating the determinant, the trace, etc, of a matrix obtained $fmm\phi_{n}(\beta)$ REFERENCES [1] J W Alexander, A lemma on systems of knotted curves, Proc Nat Acad Sci USA, 9 (1923), [2] M Arik, F Aydin, E Hizel, J Kornfilt, A Yildiz, Braid group related algebras, thier representations and genemlized hydrogenlike spectm, J Math Phys 35 (1994), [3] E Artin, Theoree der Zopfe, Abh Math Sem Univ Hamburg, 4 (1925), 47-72; Theory of bmids, Ann of Math (2) 48 (1947), [4] A Bartholomew, R Fenn, Biquandles of small size and some invariants of virtual and welded knots, arxiv: [5] S Bigelow, The Bumu representation is not faithful for n $=5$, Geometry& Topology 3 (1999), [6] J S Birman, Braids, links and mapping class groups, Annals of Mathmatics Studies, No 82 Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1974; Errata: Canad J Math 34 (1982), [7] W Burau, \"Uber Zopfgruppen und gleichsinnig verdrillte Verkettungen Abh Math Sem Hamburg 11 (1936), [8] D D Long, M Paton, The Burau representation is not faithful for n $\geq 6$, Topology 32 (1993), [9] W Magnus, A Peluso, On a theorem of V I Amol d, Comm Pure Appl Math 22 (1969), [10] A A Markov, \"Uber die freie aquivalenz der geschlossenen Zopfe, Recueil de la Soc Math de Moscou 43 (1936), [11] H Matsuda, An extension of Bumu representation, and a defomation of Alexander polynomial, preprint [12] J A Moody, The Burau representation of the bmid group $B_{n}$ is unfaithful for large n, Bull Amer Math Soc 25 (1991), [13] N Weinberg, Sur l equivalence libre des tresses fermees, Comptes Rendus (Doklady) de l Academie des Sciences de 1 URSS 23 (1939), , DEPARTMENT OF MATHEMATICS, GRADUATE SCHOOL OF SCIENCE, HIROSHIMA UNIVERSITY, HIROSHIMA , JAPAN

数理解析研究所講究録別冊 = RIMS Kokyuroku Bessa (2012), B32:

Imaginary quadratic fields whose ex Titleequal to two, II (Algebraic Number 010) Author(s) SHIMIZU, Kenichi Citation 数理解析研究所講究録別冊 = RIMS Kokyuroku Bessa (01), B3: 55-69 Issue Date 01-07 URL http://hdl.handle.net/33/19638