Narrow Results

Taking advantage of a numerical invariant, we visualize a distribution of rational homology 3-spheres on a plane via the Casson–Walker–Lescop (CWL) invariant and observe several aspects of the distribution. In particular, we study the characteristics of the distribution of lens spaces as a fundamental family of rational homology 3-spheres with a way to yield a family of estimation for the Dedekind sum. The CWL invariant captures the finiteness of lens space surgeries along knots. According to the finiteness, for example, the CWL invariant determines possible lens spaces as the results of integral surgeries along a knot $K$ with $a_1(K)=12$.

We calculate the Ohtsuki invariants ${\lambda }_i(M)$$(i=0,1,2,3)$ of every Brieskorn–Hamm manifold $M$ which is a rational homology $3$-sphere. We denote the order of $H_1(M;\mathbb{Z})$ by $H$. By the result, we show that the third Ohtsuki invariant ${\lambda }_3(M)$ of the Brieskorn–Hamm manifolds $M$ with $H=1$ is negative, and the third Ohtsuki invariant ${\lambda }_3(M)$ of most Brieskorn–Hamm manifolds $M$ with $H\ge 2$ is positive.

Let $p$ and $q$ be pairwise coprime with $p\ne 0$ and $q>0$. Let the Brieskorn–Hamm manifold be a homology lens space and the order its the first homology group is $\left|p\right|$. We classify the Brieskorn–Hamm manifold and the resulting $3$-manifold of $p/q$-surgery along a knot $K$ in $S^3$ by using the Lescop invariant.

We introduce the inversion polynomial for Dedekind sums f_b(x) = ∑ x^{inv(a, b)} to study the number of s(a, b) which have the same value for a given b. We prove several properties of this polynomial and present some conjectures. We also introduce connections between Kloosterman sums and the inversion polynomial evaluated at particular roots of unity. Finally, we improve on previously known bounds for the second highest value of the Dedekind sum and provide a conjecture for a possible generalization. Lastly, we include a new sufficient condition for the inequality of two Dedekind sums based on the reciprocity formula.

In this article, we will give a new proof of the reciprocity law for Dedekind sums, as well as a proof of the transformation formula for the Dedekind $\eta$-function using the Chinese Remainder Theorem.