On a curvature flow in a band domain with unbounded boundary slopes

<p style='text-indent:20px;'>This paper is devoted to an anisotropic curvature flow of the form <inline-formula><tex-math id="M1">\begin{document}$ V = A(\mathbf{n})H + B(\mathbf{n}) $\end{document}</tex-math></inline-formula> in a band domain <inline-formula><tex-math id="M2">\begin{document}$ \Omega : = [-1,1]\times {\mathbb{R}} $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M3">\begin{document}$ \mathbf{n} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ V $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ H $\end{document}</tex-math></inline-formula> denote respectively the unit normal vector, normal velocity and curvature of a graphic curve <inline-formula><tex-math id="M6">\begin{document}$ \Gamma_t $\end{document}</tex-math></inline-formula>. We require that the curve <inline-formula><tex-math id="M7">\begin{document}$ \Gamma_t $\end{document}</tex-math></inline-formula> contacts <inline-formula><tex-math id="M8">\begin{document}$ \partial \Omega $\end{document}</tex-math></inline-formula> with slopes equaling to the heights of the contact points (which corresponds to a kind of Robin boundary conditions). In spite of the unboundedness of the boundary slopes, we are able to obtain the <i>uniform interior gradient estimates</i> for the solutions by using the zero number argument. Furthermore, when <inline-formula><tex-math id="M9">\begin{document}$ t\to \infty $\end{document}</tex-math></inline-formula>, we show that <inline-formula><tex-math id="M10">\begin{document}$ \Gamma_t $\end{document}</tex-math></inline-formula> converges to a traveling wave with cup-shaped profile and <i>infinite</i> boundary slopes in the <inline-formula><tex-math id="M11">\begin{document}$ C^{2,1}_{\rm{loc}} ((-1,1)\times {\mathbb{R}}) $\end{document}</tex-math></inline-formula>-topology.</p>

The slope conjecture for Montesinos knots

The slope conjecture relates the degree of the colored Jones polynomial of a knot to boundary slopes of essential surfaces. We develop a general approach that matches a state-sum formula for the colored Jones polynomial with the parameters that describe surfaces in the complement. We apply this to Montesinos knots proving the slope conjecture for Montesinos knots, with some restrictions.

State invariants of 2-bridge knots

In this paper, we consider generalizations of the Alexander polynomial and signature of 2-bridge knots by considering the Gordon–Litherland bilinear forms associated with essential state surfaces of the 2-bridge knots. We show that the resulting invariants are well-defined and explore properties of these invariants. Finally, we realize the boundary slopes of essential surfaces as differences of signatures of the knot.

The slope conjecture for graph knots

The slope conjecture proposed by Garoufalidis asserts that the Jones slopes given by the sequence of degrees of the coloured Jones polynomials are boundary slopes. We verify the slope conjecture for graph knots, i.e. knots whose Gromov volume vanish.

A slope conjecture for links

The slope conjecture [S. Garoufalidis, The degree of a q-holonomic sequence is a quadratic quasi-polynomial, Electron. J. Combin. 18 (2011) 4–27] gives a precise relation between the degree of the colored Jones polynomial of a knot and the boundary slopes of essential surfaces in the knot complement. In this paper, we propose a generalization of the slope conjecture to links. We prove the conjecture for all alternating or more generally adequate links. We also verify the conjecture for torus links.

Repeated boundary slopes for 2-bridge knots

We investigate the question of when distinct branched surfaces in the complement of a 2-bridge knot support essential surfaces with identical boundary slopes. We determine all instances in which this occurs and identify an infinite family of knots for which no boundary slopes are repeated.

COEXISTENCE OF COILED SURFACES AND SPANNING SURFACES FOR KNOTS AND LINKS

It is a well-known procedure for constructing a torus knot or link that first we prepare an unknotted torus and meridian disks in its complementary solid tori, and second we smooth the intersections of the boundaries of the meridian disks uniformly. Then we obtain a torus knot or link on the unknotted torus and its Seifert surface made of meridian disks. In the present paper, we generalize this procedure by using a closed fake surface and show that the two resulting surfaces obtained by smoothing triple points uniformly are essential. We also show that a knot obtained by this procedure satisfies the Neuwirth conjecture and that the distance of two boundary slopes for the knot is equal to the number of triple points of the closed fake surface.