State graphs and fibered state surfaces

Girão, Darlan¹ ; Purcell, Jessica²

¹Department of Mathematics, Universidade Federal do Ceará, Fortaleza-CE, Brazil
²School of Mathematical Sciences, Monash University, Clayton, VIC, Australia

Algebraic and Geometric Topology, Tome 20 (2020) no. 2, pp. 987-1014

Résumé

Associated to every state surface for a knot or link is a state graph, which embeds as a spine of the state surface. A state graph can be decomposed along cut-vertices into graphs with induced planar embeddings. Associated with each such planar graph is a checkerboard surface, and each state surface is a fiber if and only if all of its associated checkerboard surfaces are fibers. We give an algebraic condition that characterizes which checkerboard surfaces are fibers directly from their state graphs. We use this to classify fibering of checkerboard surfaces for several families of planar graphs, including those associated with 2–bridge links. This characterizes fibering for many families of state surfaces.

Détail

DOI : 10.2140/agt.2020.20.987

Classification : 57M25, 57M27
Keywords: fibered links, Kauffman state, state graph, state surface, $2$–bridge link

Bibliographie

[1] P Bartholomew, S Mcquarrie, J S Purcell, K Weser, Volume and geometry of homogeneously adequate knots, J. Knot Theory Ramifications 24 (2015) | DOI

[2] S D Burton, E Kalfagianni, Geometric estimates from spanning surfaces, Bull. Lond. Math. Soc. 49 (2017) 694 | DOI

[3] R H Crowell, H F Trotter, A class of pretzel knots, Duke Math. J. 30 (1963) 373 | DOI

[4] O T Dasbach, D Futer, E Kalfagianni, X S Lin, N W Stoltzfus, The Jones polynomial and graphs on surfaces, J. Combin. Theory Ser. B 98 (2008) 384 | DOI

[5] K Finlinson, J S Purcell, Volumes of Montesinos links, Pacific J. Math. 282 (2016) 63 | DOI

[6] D Futer, Fiber detection for state surfaces, Algebr. Geom. Topol. 13 (2013) 2799 | DOI

[7] D Futer, E Kalfagianni, J Purcell, Guts of surfaces and the colored Jones polynomial, 2069, Springer (2013) | DOI

[8] D Futer, E Kalfagianni, J S Purcell, Quasifuchsian state surfaces, Trans. Amer. Math. Soc. 366 (2014) 4323 | DOI

[9] D Futer, E Kalfagianni, J S Purcell, Hyperbolic semi-adequate links, Comm. Anal. Geom. 23 (2015) 993 | DOI

[10] D Gabai, Detecting fibred links in S3, Comment. Math. Helv. 61 (1986) 519 | DOI

[11] P Ghiggini, Knot Floer homology detects genus-one fibred knots, Amer. J. Math. 130 (2008) 1151 | DOI

[12] D Girão, On the fibration of augmented link complements, Geom. Dedicata 168 (2014) 207 | DOI

[13] D Girão, J Nogueira, A Salgueiro, Fiber surfaces from alternating states, Algebr. Geom. Topol. 15 (2015) 2805 | DOI

[14] S Goodman, G Tavares, Pretzel-fibered links, Bol. Soc. Brasil. Mat. 15 (1984) 85 | DOI

[15] J Harer, How to construct all fibered knots and links, Topology 21 (1982) 263 | DOI

[16] A Hatcher, W Thurston, Incompressible surfaces in 2–bridge knot complements, Invent. Math. 79 (1985) 225 | DOI

[17] M Hirasawa, K Murasugi, Genera and fibredness of Montesinos knots, Pacific J. Math. 225 (2006) 53 | DOI

[18] E Kalfagianni, C R S Lee, On the degree of the colored Jones polynomial, Acta Math. Vietnam. 39 (2014) 549 | DOI

[19] T Kanenobu, The augmentation subgroup of a pretzel link, Math. Sem. Notes Kobe Univ. 7 (1979) 363

[20] I Kapovich, A Myasnikov, Stallings foldings and subgroups of free groups, J. Algebra 248 (2002) 608 | DOI

[21] L H Kauffman, State models and the Jones polynomial, Topology 26 (1987) 395 | DOI

[22] K Murasugi, On a certain subgroup of the group of an alternating link, Amer. J. Math. 85 (1963) 544 | DOI

[23] K Murasugi, Knot theory & its applications, Birkhäuser (2008) | DOI

[24] L Neuwirth, Knot groups, PhD thesis, Princeton University (1959)

[25] Y Ni, Knot Floer homology detects fibred knots, Invent. Math. 170 (2007) 577 | DOI

[26] M Ozawa, Essential state surfaces for knots and links, J. Aust. Math. Soc. 91 (2011) 391 | DOI

[27] M D Pabiniak, J H Przytycki, R Sazdanović, On the first group of the chromatic cohomology of graphs, Geom. Dedicata 140 (2009) 19 | DOI

[28] R L Parris, Pretzel knots, PhD thesis, Princeton University (1978)

[29] J Stallings, On fibering certain 3–manifolds, from: "Topology of –manifolds and related topics" (editor J M. K. Fort), Prentice-Hall (1962) 95

[30] J R Stallings, Constructions of fibred knots and links, from: "Algebraic and geometric topology, II" (editor R J Milgram), Proc. Sympos. Pure Math. 32, Amer. Math. Soc. (1978) 55

[31] J R Stallings, Topology of finite graphs, Invent. Math. 71 (1983) 551 | DOI

[32] N W M Touikan, A fast algorithm for Stallings’ folding process, Internat. J. Algebra Comput. 16 (2006) 1031 | DOI