By C. Stanley Ogilvy

Subject matters together with harmonic department and Apollonian circles, inversive geometry, the hexlet, conic sections, projective geometry, the Golden part and attitude trisection are addressed in a manner that brings out the genuine highbrow pleasure inherent in every one. additionally incorporated: a few unsolved difficulties of recent geometry. Notes. References. 132 line illustrations.

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Low Dimensional Topology

During this quantity, that is devoted to H. Seifert, are papers in response to talks given on the Isle of Thorns convention on low dimensional topology held in 1982.

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2Cn ], Ci = 0. Here S(p, q) is a link if and only if the length n is odd. 3. The braid index of 2-bridge links Let L be an oriented 2-bridge knot or link. In 1991, K. Murasugi [9] determined the braid index of L, using Morton-Franks-Williams inequality [8, 4] and Yamada’s braiding algorithm [12]. 4 of P. 1 (Murasugi). Let L be a 2-bridge link S(p, q), where 0 < q < p and q is odd. Let [2C1,1 , 2C1,2 , . . , 2C1,k1 , −2C2,1 , −2C2,2 , . . , −2C2,k2 , . . , (−1)t−1 2Ct,1 , . . , (−1)t−1 2Ct,kt ] be the unique continued fraction for q/p, where Ci,j > 0 for all i, j.

M. Khovanov, Categorification of the Jones polynomial. Duke Math. J. 87 (1997), 409-480. March 4, 2007 11:41 WSPC - Proceedings Trim Size: 9in x 6in ws-procs9x6 25 6. K. Murasugi. The Jones polynomials of periodic links. Paciﬁc J. Math. 131 (1988) pp. 319-329. 7. P. Traczyk. 10101 has no period 7: A criterion for periodicity of links. Proc. Amer. Math. Soc. 108, pp. 845-846. 1990. 8. O. Viro, Khovanov homology, its deﬁnitions and ramiﬁcations. Fund. Math. 184 (2004), 317-342 March 4, 2007 11:41 WSPC - Proceedings Trim Size: 9in x 6in This page intentionally left blank ws-procs9x6 March 27, 2007 11:10 WSPC - Proceedings Trim Size: 9in x 6in ws-procs9x6 Intelligence of Low Dimensional Topology 2006 Eds.

1) MN (L) = 0 if and only if L is ﬁbred. (2) MN (L) = 2 × min{m1 (f ) | f is moderate}. 2. Heegaard splitting for sutured manifolds and product decompositions We recall the deﬁnition of a sutured manifold ([1]). A sutured manifold (M, γ) is a compact oriented 3-dimensional manifold M together with a set γ(⊂ ∂M ) of mutually disjoint annuli A(γ) and tori T (γ). In this paper, we deal with the case of T (γ) = ∅. The core curve of a component of A(γ) is called a suture, and we denote by s(γ) the set of sutures.