Narrow Results

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.

We compute Vassiliev invariants up to order six for arbitrary pretzel knots, which depend on $g+1$ parameters $n_1,\dots ,n_{g+1}$. These invariants are symmetric polynomials in $n_1,\dots ,n_{g+1}$ whose degree coincide with their order. We also discuss their topological and integer-valued properties.

Let ℚ(α, s) be the field of rational functions in α, s. We compute the Kauffman polynomials of pretzel knots [1,6] using the Kauffman skein theory and linear algebra tools. We give a formula for the Kauffman polynomial of a pretzel knot such that after inputting the sequence notation of the pretzel knot, the output is its Kauffman polynomial. Our calculation can be implemented in Mathematica, Maple, Mathcad, etc.

We calculate the rational Khovanov homology of a class of pretzel knots by using the spectral sequence constructed by Turner. Moreover, we determine Rasmussen's s-invariant of almost all pretzel knots P(p, q, r) by using Turner's spectral sequence, a sharper slice-Bennequin inequality, and a skein inequality.

Conjecture $\mathbb{Z}$ is a knot theoretical equivalent form of the Kervaire conjecture. We show that Conjecture $\mathbb{Z}$ is true for all the pretzel knots of the form $P(p,q,-r)$ where $p$, $q$ and $r$ are odd positive integers.

Greene-Jabuka and Lecuona confirmed the slice-ribbon conjecture for 3-stranded pretzel knots except for an infinite family $P(a,-a-2,-\frac{{(a+1)}^2}{2})$, where $a$ is an odd integer greater than $1$. Lecuona and Miller showed that $P(a,-a-2,-\frac{{(a+1)}^2}{2})$ are not slice unless $a\equiv 1,11,37,47,59(mod60)$. In this note, we show that four-fifths of the remaining knots in the family are not slice.

We show that Vassiliev invariants of orders ≤ 10 do not detect the change of orientation in a large class of pretzel knots. We also show that for every n ∈ N there exist non-invertible knots that have the same Vassiliev invariants of orders ≤ n, with their inverses.