The strong slope conjecture for twisted generalized Whitehead doubles (Q2224470)

Let \(J_{K,n}(q)\in\mathbb{Z}[q^{\pm1/2}]\) be the colored Jones polynomial of a knot \(K\) in the three-sphere \(S^3\). It is normalized so that \(J_{U,n}(q)=(q^{n/2}-q^{-n/2})/(q^{1/2}-q^{-1/2})\) for the unknot \(U\) and that \(J_{K,2}(q)/J_{U,2}(q)\) is the original Jones polynomial [\textit{V. F. R. Jones}, Bull. Am. Math. Soc., New Ser. 12, 103--111 (1985; Zbl 0564.57006)]. It is known that for sufficiently large \(n\), its maximum degree is a quadratic quasi-polynomial \(a(n)n^2+b(n)n+c(n)\) for periodic functions \(a(n)\), \(b(n)\), and \(c(n)\) with integral periods [\textit{S. Garoufalidis} and \textit{T. T. Q. Lê}, Geom. Topol. 9, 1253--1293 (2005; Zbl 1078.57012); \textit{S. Garoufalidis}, Electron. J. Comb. 18, No. 2, Research Paper P4, 23 p. (2011; Zbl 1298.05042)]. A Jones slope of \(K\) is defined to be \(4a(n)\) for some \(n\). \par An element \(p/q\in\mathbb{Q}\cup\{1/0\}\) is called a boundary slope of a knot \(K\subset S^3\) if there exists an essential surface in \(S^3\setminus\operatorname{Int}{N(K)}\) with a boundary component presenting \(p\mu+q\lambda\in H_1(\partial{N(K)},\mathbb{Z})\), where \(N(K)\) is the closed tubular neighborhood of \(K\), and \((\mu,\lambda)\) is the standard meridian-longitude basis of \(H_1(\partial{N(K)};\mathbb{Z})\). Here a properly embedded surface in a three-manifold is called essential if each of its components is orientable, incompressible, and boundary-incompressible. \par The Slope Conjecture states that a Jones slope is a boundary slope for any knot [\textit{S. Garoufalidis}, Quantum Topol. 2, No. 1, 43--69 (2011; Zbl 1228.57004)]. The Strong Slope Conjecture states that for any knot \(K\) and any Jones slope \(p/q\) (\(q>0\)) of \(K\), there exists an essential surface \(F\subset S^3\setminus\operatorname{Int}{N(K)}\) with \(|\partial{F}|\) boundary components such that each component of \(\partial{F}\) has slope \(p/q\) and \(2b(n)=\frac{\chi(F)}{q|\partial F|}\) for some \(n\), where \(\chi\) denotes the Euler characteristic [\textit{E. Kalfagianni} and \textit{A. T. Tran}, New York J. Math. 21, 905--941 (2015; Zbl 1331.57022)]. Let \(k\cup\ell\) be the negative Whitehead link and \(k^{0}_{1}\) be the component \(k\) regarded as a knot in \(S^3\setminus{N(\ell)}\). Note that \(S^3\setminus{N(\ell)}\) is the interior of a solid torus \(V\) and so we assume that it is standardly embedded in \(S^3\). Now, let \(k^{\tau}_{\omega}\subset\operatorname{Int}{V}\) be a knot obtained from \(k^{0}_{1}\subset\operatorname{Int}{V}\) by giving \(\tau\) full-twists around the meridian and adding \(\omega-1\) full-twists around the clasp. The \(\tau\)-twisted \(\omega\)-generalized Whitehead double \(W^{\tau}_{\omega}(K)\) of a knot \(K\) is defined as the satellite knot with companion \(K\) and pattern \((V,k^{\tau}_{\omega})\). Note that \(K^{\tau}_{0}\) is just the unknot for every \(\tau\). \par In the paper under review, the authors prove (1) under certain assumptions if the Slope Conjecture is true for \(K\), then so it is for \(W^{\tau}_{\omega}(K)\) for every pair \((\tau,\omega)\) with \(\omega\ne0\), and (2) if the Strong Slope Conjecture is true for \(K\), then the Strong Slope Conjecture is true for \(W^{\tau}_{\omega}(K)\) for every pair \((\tau,\omega)\) with \(\omega\ne0\). \par The proof depends on concrete calculations of the colored Jones polynomials by using skein theory [\textit{G. Masbaum} and \textit{P. Vogel}, Pac. J. Math. 164, No. 2, 361--381 (1994; Zbl 0838.57007)] and a careful enumeration of essential surfaces in \(V\setminus\operatorname{Int}{N(k^{\tau}_{\omega})}\) following [\textit{W. Floyd} and \textit{A. E. Hatcher}, Trans. Am. Math. Soc. 305, No. 2, 575--599 (1988; Zbl 0672.57006)] and [\textit{J. Hoste} and \textit{P. D. Shanahan}, Math. Comput. 76, No. 259, 1521--1545 (2007; Zbl 1137.57005)]. Note that \(V\setminus\operatorname{Int}{N(k^{\tau}_{\omega})}\) is homeomorphic to the complement of the two-bridge link \([2,2\omega,-2]\) in the Conway notation. Together with the results in [\textit{S. Garoufalidis} and \textit{T. T. Q. Lê}, loc. cit.; \textit{D. Futer} et al., Proc. Am. Math. Soc. 139, No. 5, 1889--1896 (2011; Zbl 1232.57007); \textit{E. Kalfagianni} and \textit{A. T. Tran}, loc. cit.; \textit{K. L. Baker} et al., New York J. Math. 27, 676--704 (2021; Zbl 1485.57002)], it follows that the Strong Slope Conjecture (and hence the Slope Conjecture) is true for any knot obtained by a finite sequence of cablings, untwisted \(\omega\)-generalized Whitehead doublings (\(\omega>0\)), and connected-sums of \(B\)-adequate knots, such as alternating knots, or torus knots.