Remarks on Jones Slopes and Surfaces of Knots

Acta Mathematica Vietnamica - Tập 46 - Trang 289-299 - 2021
Efstratia Kalfagianni1
1Department of Mathematics, Michigan State University, East Lansing, USA

Tóm tắt

We show that the strong slope conjecture implies that the degrees of the colored Jones knot polynomials detect the figure 8 knot. Furthermore, we propose a characterization of alternating knots in terms of the Jones period and the degree span of the colored Jones polynomial

Tài liệu tham khảo

Baker, K.L., Motegi, K., Takata, T.: The strong slope conjecture for graph knots. arXiv:1809.01039 (2018) Baker, K.L., Motegi, K., Takata, T.: The strong slope conjecture for twisted generalized Whitehead doubles. Quantum Topol. 11(3), 545–608 (2020) Dasbach, O.T., Futer, D., Kalfagianni, E., Lin, X.S., Stoltzfus, lN.W.: The Jones polynomial and graphs on surfaces. J. Combin. Theory Ser. B 98(2), 384–399 (2008) Futer, D., Kalfagianni, E., Purcell, J.S.: Slopes and colored Jones polynomials of adequate knots. Proc. Am. Math. Soc. 139, 1889–1896 (2011) Futer, D., Kalfagianni, E., Purcell, J.S.: Guts of Surfaces and the Colored Jones Polynomial Lecture Notes in Mathematics, vol. 2069. Springer, Heidelberg (2013) Futer, D., Kalfagianni, E., Purcell, J.S.: Jones polynomials, volume and essential knot surfaces: a survey. In: Knots in Poland. III. Part 1. Banach Center Publ., vol. 100, 51–77. Polish Acad. Sci. Inst. Math., Warsaw (2014) Garoufalidis, S.: The degree of a q-holonomic sequence is a quadratic quasi-polynomial. Electron. J. Combin. 18(2), paper 4, 23 (2011) Garoufalidis, S.: The Jones slopes of a knot. Quantum Topol. 2(1), 43–69 (2011) Garoufalidis, S., Lee, C.R.S., van der Veen, R.: The slope conjecture for Montesinos knots. Internat. J. Math. 31(7), 2050056, 66 (2020) Gordon, C.McA: Boundary slopes of punctured tori in 3-manifolds. Trans. Am. Math. Soc. 350(5), 1713–1790 (1998) Gordon, C.McA., Luecke, J.: Knots are determined by their complements. Bull. Am. Math. Soc. (N.S.) 20(1), 83–87 (1989) Hatcher, A.E.: On the boundary curves of incompressible surfaces. Pacific J. Math. 99(2), 373–377 (1982) Howie, J.A.: Coiled surfaces and the strong slope conjecture. Preprint Howie, J.A.: A characterisation of alternating knot exteriors. Geom. Topol. 21(4), 2353–2371 (2017) Kalfagianni, E.: A Jones slopes characterization of adequate knots. Indiana Univ. Math. J. 67(1), 205–219 (2018) Kalfagianni, E.: The strong slope conjecture and torus knots. J. Math. Soc. Japan 72(1), 73–79 (2020) Kalfagianni, E., Lee, C.R.S.: Normal and Jones surfaces of knots. J. Knot Theory Ramifications 27(6), 1850039, 15 (2018) Kalfagianni, E., Tran, A.T.: Knot cabling and degrees of colored Jones polynomials. New York J. Math. 21, 905–941 (2015) Lackenby, M., Meyerhoff, R.: The maximal number of exceptional Dehn surgeries. Invent. Math. 191(2), 341–382 (2013) Lee, C.R.S.: Jones slopes and coarse volume for near-alternating links. Comm. Anal. Geom. (To appear) Lee, C.R.S., van der Veen, R.: Cancellations in the degree of the colored Jones polynomial. arXiv:2006.01303 Lee, C.R.S., van der Veen, R.: Slopes for pretzel knots. New York J. Math. 22, 1339–1364 (2016) Leng, X., Yang, Z., Liu, X.: The slope conjectures for 3-string Montesinos knots. New York J. Math. 25, 45–70 (2019) Lickorish, W.B.R.: An Introduction to Knot Theory Graduate Texts in Mathematics, vol. 175. Springer, New York (1997) Lickorish, W.B.R., Thistlethwaite, M.B.: Some links with nontrivial polynomials and their crossing-numbers. Comment. Math. Helv. 63(4), 527–539 (1988) Motegi, K., Takata, T.: The slope conjecture for graph knots. Math. Proc. Cambridge Philos. Soc. 162(3), 383–392 (2017) Ozawa, M.: Essential state surfaces for knots and links. J. Aust. Math. Soc. 91(3), 391–404 (2011)