第 61 回トポロジーシンポジウム講演集 2014 年 7 月於東北大学 ( ) 1 ( ) [6],[7] J.W. Alexander 3 1 : t 2 t +1=0 4 1 : t 2 3t +1=0 8 2 : 1 3t +3t 2 3t 3 +3t 4 3t 5 + t

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1 ( ) 1 ( ) [6],[7] J.W. Alexander 3 1 : t 2 t +1=0 4 1 : t 2 3t +1=0 8 2 : 1 3t +3t 2 3t 3 +3t 4 3t 5 + t : 1 5t +9t 2 5t 3 + t 4 ( : ) 2010 Mathematics Subject Classification: 57M25, 57M27, 26C10 knots and links, Alexander polynomial, distribution of zeros hirasawa.mikami@nitech.ac.jp 27

2 α ᾱ α α ᾱ 1/α 1/ᾱ α 1/α L Δ L (t) 2.1 L Δ L (t) (1) real stable (r-stable). (2) circular stable (c-stable). (3) bi-stable. stability stability D.G.Wagner [16] ) H C n f(z 1,...,z n ) C[z 1,...,z n ] H {α 1,...,α n } f(α 1,...,α n ) 0 H-stable H {α C Re(α) > 0} f Hurwitz-stable H {α C Im(α) > 0}, f stable polynomial f real stable f f real stable f f Hurwitz-stable f α Re(α) 0 < strongly Hurwitz-stable strongly Hurwitz-stability Hoste (2002). K 1 Δ K ( (t +1)) strongly Hurwitz-stable 2.2 M f (i) f strongly Hurwitz-stable. (ii) V,W VM+ M T V = W. [10] Hoste stability 2.3 [10] ( α 3 < Re(α) < 6 28

2.2. Hoste 1 3.1 I.D.Jong 2 [3]. Hartley [4] -Suketa [13].

3 2.2. Hoste I.D.Jong 2 [3]. Hartley [4] -Suketa [13]. 20 K =[2, 2,, 2, 2, 2,, 2]. n }{{}}{{} n n real stable c-stable 3 stable. real stable c-stable 29

4 2.3. Δ K (t) = 2n j=0 ( 1)j c j t 2n j. c 0 <c 1 < <c k = c k+1 = = c 2n k >c 2n k+1 >c 2n [4] [12] (unimodal) [17] real stable [5] real stable K σ(k) 0 [11] σ(k) ±2 1 σ(k) c-stable 2.5. bi-orderability real stability biorderability Clay-Rolfsen [2] G leftorderable G f,g,h f <g hf < hg h G bi-orderable Rolfsen left-orderable K K real stable G(K) bi-orderable [14] G(K) bi-orderable Δ K (t) [2]. K bi-orderable [2]. real stable 30

5 =[2, 2], 4 1 =[2, 2] M a a a 3 1 M = 0 a 4 1 Δ K (t) =det(tm M T ) a m a m 3.1 K [2a 1, 2b 1,...,2a n, 2b n ] (i) a i,b i > 0 K c-stable. (ii) a i > 0,b i < 0 K r-stable. K strictly bi-stable bi-stable totally unstable 3.2 [7, Propositions 8.1, 8.3] a, b, c > 0 (i) [2a, 2, 2b, 2c] bc > 2a(c +1) K r-stable. (ii) [2a, 2b, 2b, 2a] a 4b K r-stable. 3.3 [7, Proposition 12.2] K =[2, 2k, 2, 2] (i) k<0 K strictly bi-stable. (ii) k =1, 2, 3 K totally unstable. (iii) k 4 K c-stable. 3.4 [7, Theorem 13.1] K =[2a 1, 2a 2,...,2a 2m, 2b 1, 2c 1,...,2b p, 2c p ], a j,b j,c j > 0 K bi-stable 2p 2m 31

6 3.2. bi-stable bi-stable 2 Mahler measure Mahler measure P ( 2, 3, 7) Mahler measure [9]). 2 2 [8] 2 [2, 2,, 2, 2, 2, 2] }{{} 5 [2, 2,, 2, 2,, 2] [2, 2,, 2, 2, 2, 2, 2], }{{}}{{}}{{} [2, 2,, 2, 2, 2, 2, 2], [2, 2,, 2, 2,, 2, 2, 2, 2] } {{ } 4 4. } {{ } real stable } {{ } 5 M Δ K (t) =det(tm M T ) = det M det(te M 1 M T ) Δ det M N = M 1 M T N K real stable M 4.1 M [ ] A B A O H M = M 1 M T H B rational knot stable quasi-rational knots (link) quasi-rational quasi-rational quasi-rational 32

7 K = X(2a 1, 2a 2,, 2a n 2b 1, 2b 2,, 2b n ). K X-type X(2 2) 4 1 X(2, 2 2, 2) 8 12 X(2, 2, 2 2, 2, 2) 12 a0125 K ±2 4.2 a j, b j X-type r-stable [10] α 3 < Re(α) < 6 X-type X(2,...,2 2,..., 2) 4.2. c-stable M K M + M T σ(m + M T ) K Milnor σ(k) #( ) σ(k) = deg Δ K (t) K c-stable quasi-rational X-type X(2, 2, 2 2, 2, 2) c-stable strictly bi-stable X(4, 2, 2, 2, 2, 2) c-stable σ(k) c-stable c-stable 4.3 F rankh 1 (F, Z) =n F n α 1,...,α n F \ i α i N F α j N F F c-stable F F reciprocal [15] 33

8 4.3. bi-stable Δ K (t) F (x) Δ K (t) t t 1/t x = t + 1 t x (z 2 ) z 2n (x 2) n : Δ=t 6 3t 5 +2t 4 t 3 +2t 2 3t +1 t 3 3t t 3 1 t t 3 = t t 3 3(t t 2 )+2(t + 1 t ) 1=(t + 1 t )3 3(t + 1 t ) 3 ( (t + 1 t )2 2 ) +2(t + 1 t ) 1=x3 3x 2 x K bi-stable F (x) r-stable. #(Δ )= 2N, N =#(F (x) α α 2 ). 3.7 [2, 2k, 2, 2] 5. interlacing property a j > 0 K =[2a 1, 2a 2,, ( 1) k 1 2a k,, ( 1) n 1 2a n ], K = [2a 1, 2a 2,, ( 1) n 2 2a n 1 ] Δ K (t)δ K (t) Δ K (t) Δ K (t) n interlacing interlace 34

9 real stable 5.2. interlacing property interlacing property 3.1 quasirational X-type Y -type Y (2a 1, 2a 2,, 2a 2n+1 2b 1, 2b 2,, 2b 2n+1 ) n =3 Y (2 2) 4 1 Y (2, 2, 2 2, 2, 2) 12 a1124. a j b j Y -type real sable a a1124 X-type Y -type interlacing property bi-stable interlacing property 6. [1] J.W. Alexander, Topological invariants of knots and links, Trans. Amer. Math. Soc. 30 (1928) [2] A.Clay and D.Rolfsen, Ordered groups, eigenvalues, knots, surgery and L-spaces, Math. Proc. Camb. Phill. Soc. 152 (2012) [3] I.D.Jong, Alexander polynomials of alternating knot of genus two II, J.Knottheory Ramifications 19 (2010) [4] R.Hartley, On two-bridged knots polynomials, J. Austral. Math. Soc. Ser A 28 (1979),

10 [5] June Huh, Milnor numbers of projective hypersurfaces and the chromatic polynomial of graphs, J. Amer. Math. Soc. 25 (2012) [6] M.Hirasawa and K.Murasugi, On stability of Alexander polynomials of knots and links (Survey), Proc. of Knots in Poland III, 2010, Banach Center Publications vol 100, 2013 Poland. [7], Various stabilities of the Alexander polynomials of knots and links, arxiv: (2013) 92pages. [8] E. Hironaka, Salem-Boyd sequences and Hopf plumbing, Osaka J. Math. 43 (2006) [9] D. Lehmer, Factorization of certain cyclotomic functions, Ann. of Math. 34 (1933), [10] L.Lyubich and K.Murasugi, On zeros of the Alexander polynomial of an alternating knot, Topology Appl. 159 (2012) [11] J.Milnor, Infinite cyclic covers, In Conf. Topology of 3-manifolds 1968 (ed.j.g.hocking), Boston-London-Sydney: Prindle, Webber and Schmdit [12] K. Murasugi, On the Alexander polynomial of alternating algebraic knots, J. Austral. Math. Soc. Ser A 39 (1985), [13] Y.Nakanishi and M.Suketa, Alexander polynomials of two-bridge knots,j.austral.math. Soc. Ser A 60 (1996) [14] B.Perron and D.Rolfsen, On orderability of fibred knot group, Math. Proc. Camb. Phill. Soc. 135 (2003) [15] M.Suzuki, An inverse problem for a class of canonical systems and its applications to self-reciprocal polynomials, arxiv: (2013) [16] D.G.Wagner, Multivariate stable polynomials: Theory and Applications, Bull. Amer. Math. Soc. 48 (2011) [17] H.S.Wilf, Generatingfunctionology, Academic Press,

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