The m-trace of a knot is the 4-manifold obtained from $B^4$ by attaching a 2-handle along the knot with m-framing. In 2015, Abe, Jong, Luecke and Osoinach introduced a technique to construct infinitely many knots with the same (diffeomorphic) m-trace, which is called the operation (∗m). In 2018, Miller and Piccirillo gave pairs of knots with diffeomorphic m-traces by utilizing Gompf and Miyazaki’s dualizable pattern. In this paper, we clarify the relation between the two techniques. In particular, we prove that the “twistings” appearing in both techniques are corresponding. In addition, we show that the family of knots admitting the same 4-surgery given by Teragaito can be explained by the operation (∗m).

In this paper, we study ${\mathbf G}_a$-actions on the affine spaces over a commutative ring of characteristic $p^e$, where $p$ is a prime number and $e\geq 2$. We say that a ${\mathbf G}_a$-action is red-nontrivial (resp. red-trivial) if it is nontrivial (resp. trivial) modulo $p$. We give a structure theorem for red-nontrivial ${\mathbf G}_a$-actions on the affine lines under some mild assumptions. Interestingly, the invariant ring for such an action is either the ring of constants or non-finitely generated. We show that every red-trivial ${\mathbf G}_a$-action on the affine space over a certain class of commutative rings is uniquely determined by two derivations, whose invariant ring is finitely generated if the base ring is noetherian. By combining these results, we completely determine the ${\mathbf G}_a$-actions on the affine lines over a certain class of commutative rings of positive characteristic, including ${\mathbf Z}/m{\mathbf Z}$ for any $m \geq 2$.

We show that the ring of modular forms with characters for the even unimodular lattice of signature (2,18) is obtained from the invariant ring of $\mathrm {Sym}(\mathrm {Sym}^8(V)\oplus \mathrm {Sym}^{12}(V))$ with respect to the action of $\mathrm{SL}(V)$ by adding a Borcherds product of weight 132 with one relation of weight 264, where $V$ is a 2-dimensional $\mathbb C$-vector space. The proof is based on the study of the moduli space of elliptic K3 surfaces with a section.

Suppose that a smooth map $(f,g,h):\mathbb{R}^n \rightarrow \mathbb{R}^3$, where $n \geq3$, has a stable singularity at the origin. We characterize the stability of the function $f:\mathbb{R}^n \rightarrow \mathbb{R}$ and the map $(f,g):\mathbb{R}^n \rightarrow \mathbb{R}^2$ at the origin in terms of the discriminant set of $(f,g,h)$.

In this work, we examine the Lyubeznik numbers of squarefree monomial ideals that are linked. Also we study these numbers for almost complete intersection ideals.

We study the ring extensions $R \subseteq T$ having the same set of prime ideals provided $\mathrm{Nil}(R)$ is a divided prime ideal. Some conditions are given under which no such $T$ exists properly containing $R$. Using idealization theory, the examples are also discussed to strengthen the results.

In an interesting article entitled “Experiments and discoveries in $q$-trigonometry”, R. W. Gosper conjectured few beautiful $\Pi_q$ and Lambert series identities. Many people have attempted confirming some of those identities in the Gosper’s list, mainly by using Gosper’s $q$-trigonometric identities. In this paper we either prove or disprove all the $\Pi_q$ and Lambert series identities in the Gosper’s list by mainly using S. Ramanujan’s theta function identities and W. N. Bailey’s summation formula. In the process, we obtain three new Gosper kind of identities.