对旋结桥分解的Kirby-Thompson不变量的界

Some lower bounds for the Kirby-Thompson invariant

Kirby and Thompson introduced a non-negative integer-valued invariant, called the Kirby-Thompson invariant, of a $4$-manifold using trisections. In this paper, we give some lower bounds for the Kirby-Thompson invariant of certain $4$-manifolds. As an application, we determine the Kirby-Thompson invariant of the spin of $L(2,1)$, which is the first example of a $4$-manifold with non-trivial Kirby-Thompson invariant. We also show that there exist $4$-manifolds with arbitrarily large Kirby-Thompson invariant.

Trisections with Kirby-Thompson length 2

Kirby and Thompson introduced a length of a trisection. They also defined the length of a 4-manifold as the minimum of length among all lengths of trisection of a 4-manifold. In this paper, we consider trisections whose Kirby-Thompson length is 2. Kirby and Thompson conjectured that length 2 trisection is a trisection of 4-manifold with length 0. We shall prove this conjecture in this paper.

结表面的Kirby-Thompson不变量的界

Bounds for Kirby-Thompson invariants of knotted surfaces

本文研究结表面的Kirby-Thompson不变量，通过两种不同方法计算其精确值，并给出下界。

Kirby-Thompson distance for trisections of knotted surfaces

We adapt work of Kirby-Thompson and Zupan to define an integer invariant $\mathcal{L}(\mathcal{T})$ of a bridge trisection $\mathcal{T}$ of a smooth surface $\mathcal{K}$ in $S^4$ or $B^4$. We show that when $\mathcal{L}(\mathcal{T})=0$, then the surface $\mathcal{K}$ is unknotted. We also show show that for a trisection $\mathcal{T}$ of an irreducible surface, bridge number produces a lower bound for $\mathcal{L}(\mathcal{T})$. Consequently, $\mathcal{L}$ can be arbitrarily large.

估算吉川表中曲面链的Kirby--Thompson不变量

Estimating the Kirby--Thompson Invariants of Surface-Links in the Yoshikawa Table

针对吉川表中的曲面链，本文借助已有研究算出的极小三平面图，推导得到其中若干曲面链的L及L*型Kirby--Thompson不变量的上界。

Constructing a broken Lefschetz fibration of S^4 with a spun or twist-spun torus knot fiber

Much work has been done on the existence and uniqueness of broken Lefschetz fibrations such as those by Auroux et al., Gay and Kirby, Lekili, Akbulut and Karakurt, Baykur, and Williams, but there has been a lack of explicit examples. A theorem of Gay and Kirby suggests the existence of a broken Lefschetz fibration of S^4 over S^2 with a 2-knot fiber. In the case of a spun or twist-spun torus knot, we present a procedure to construct such fibrations explicitly. The fibrations constructed have no cusps nor Lefschetz singularities.