# The basket numbers of knots

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[1] J. H. Conway, An enumeration of knots and links and some of their algebraic properties, Proceedings of the conference on Computational problems in Ab- stract Algebra held at Oxford in 1967, Pergamon Press, 329–358. Google Scholar

[2] Y. Choi, Y. Chung and D. Kim, The complete list of prime knots whose flat plumbing basket numbers are 6 or less, preprint, arXiv:1408.3729. Google Scholar

[3] R. Furihata, M. Hirasawa and T. Kobayashi, Seifert surfaces in open books, and a new coding algorithm for links, Bull. London Math. Soc. 40 (3) (2008), 405–414. Google Scholar

[4] D. Gabai, The Murasugi sum is a natural geometric operation, in: Low- Dimensional Topology (San Francisco, CA, USA, 1981), Amer. Math. Soc., Providence, RI, 1983, 131–143. Google Scholar

[5] D. Gabai, The Murasugi sum is a natural geometric operation II, in: Combi- natorial Methods in Topology and Algebraic Geometry (Rochester, NY, USA, 1982), Amer. Math. Soc., Providence, RI, 1985, 93–100. Google Scholar

[6] J. Gross, D. Robbins and T. Tucker, Genus distribution for bouquets of circles, J. Combin. Theory B. Soc. 47 (3) (1989) 292–306. Google Scholar

[7] J. Gross and T. Tucker, Topological graph theory, Wiley-Interscience Series in discrete Mathematics and Optimization, Wiley & Sons, New York, 1987. Google Scholar

[8] J. Harer, How to construct all fibered knots and links, Topology 21 (3) (1982) 263–280. Google Scholar

[9] S. Hirose and Y. Nakashima, Seifert surfaces in open books, and pass moves on links, arXiv:1311.3383. Google Scholar

[10] C. Hayashi and M. Wada, Constructing links by plumbing flat annuli, J. Knot Theory Ramifications 2 (1993), 427–429. Google Scholar

[11] D. Kim, Basket, flat plumbing and flat plumbing basket surfaces derived from induced graphs, preprint, arXiv:1108.1455. Google Scholar

[12] D. Kim, The boundaries of dipole graphs and the complete bipartite graphs K2,n, Honam. Math. J. 36 (2) (2014), 399–415, arXiv:1302.3829. Google Scholar

[13] D. Kim, A classification of links of the flat plumbing basket numbers 4 or less, Korean J. of Math. 22 (2) (2014), 253–264. Google Scholar

[14] L. H. Kauffman and S. Lambropoulou, On the Classification of Rational Knots, Adv. Appl. Math. 33 (2) (2004), 199–237. Google Scholar

[15] D. Kim, Y. S. Kwon and J. Lee, Banded surfaces, banded links, band indices and genera of links, J. Knot Theory Ramifications 22(7) 1350035 (2013), 1–18, arXiv:1105.0059. Google Scholar

[16] T. Nakamura, On canonical genus of fibered knot, J. Knot Theory Ramifications 11 (2002), 341–352. Google Scholar

[17] T. Nakamura, Notes on braidzel surfaces for links, Proc. of AMS 135 (2) (2007), 559–567. Google Scholar

[18] K. Reidemeister, Homotopieringe und Linsenr ̈aume, Abh. Math. Sem. Hansis- chen Univ., 11 (1936), 102–109. Google Scholar

[19] L. Rudolph, Quasipositive annuli (Constructions of quasipositive knots and links IV.), J. Knot Theory Ramifications 1 (4) (1992) 451–466.. Google Scholar

[20] L. Rudolph, Hopf plumbing, arborescent Seifert surfaces, baskets, espaliers, and homogeneous braids, Topology Appl. 116 (2001), 255–277. Google Scholar

[21] H. Schubert, Knoten mit zwei Bru ̈cken, Math. Zeitschrift, 65 (1956), 133–170. Google Scholar

[22] H. Seifert, Uber das Geschlecht von Knoten, Math. Ann. 110 (1934), 571–592. Google Scholar

[23] J. Stallings, Constructions of fibred knots and links, in: Algebraic and Geometric Topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, CA, 1976), Part 2, Amer. Math. Soc., Providence, RI, 1978, pp. 55–60. Google Scholar

[24] M. Thistlethwaite, Knotscape, available at http://www.math.utk.edu/∼morwen/knotscape. html. Google Scholar

[25] T. Van Zandt. PSTricks: PostScript macros for generic TEX. Available at ftp://ftp. princeton.edu/pub/tvz/. Google Scholar