Let M be a compact, orientable 3-manifold and F an essential closed surface which cuts M into M 1 and M 2 . Suppose that M i has a Heegaard splitting, and the amalgamation of V 1 ∪ S 1 W 1 and V 2 ∪ S 2 W 2 is the unique minimal Heegaard splitting of M up to isotopy.

Abstract. In this paper, we investigate the simplicial groups obtained from the link groups of naive cablings on any given framed link. Our main result states that the resulting simplicial groups have the homotopy type of the loop space of a wedge of 3-spheres. This gives simplicial group models for some loop spaces using link groups.

A graph embedded in the 3-sphere is called irreducible if it is non-splittable and for any 2-sphere embedded in the 3-sphere that intersects the graph at one point the graph is contained in one of the 3-balls bounded by the 2-sphere. We show that irreducibility is preserved under certain deformations of embedded graphs. We show that certain embedded graphs are irreducible.

In a 21st century context characterized by the inefficient use of urban construction land, to optimize land use structure and transform resource utilization methods, it is necessary for the Chinese government to improve land use efficiency. Promoting the readjustment of inefficient urban land use has become essential. The purpose of this paper is to sort through the policies addressing the readjustment of inefficient urban land use in China, identify the promulgation date, longitudinal hierarchy, and horizontal composition of the relevant policies, and summarize the evolutionary trend of such policies. This study aims to provide a reference for the adjustment and improvement of relevant policy systems in China. The content analysis method was used in this study, and NVivo 12 software was used to compare and verify the analysis process. The results show the following: (1) A large number of relevant policies have been promulgated, with a total of 12 items from 1988 to 2012, increasing to six items annually on average from 2013 to 2019. The Communist Party of China Central Committee, the National People’s Congress of the People’s Republic, the State Council and relevant ministries promulgated 13.46%, 11.54%, 28.85%, and 46.15%, respectively, of the relevant policies. (2) The existing policies are mostly issued in the form of notices, opinions, management regulations, etc.; therefore, their level of effectiveness is low, and the role of norms is weak. (3) Finally, the policy content experienced stages of initial exploration, basic establishment, and continuous improvement. It also covered aspects such as Standard Control, Market Configuration, Overall Planning, Incentive Mechanisms, Income Distribution, and Supervision Evaluation. The system for policies addressing readjustment of inefficient urban land use has basically been perfected, although post approval supervision still needs to be strengthened. Therefore, the process of readjusting inefficient urban land use should be optimized, new modes of readjustment should be constantly explored, and inefficient land use should be addressed across the entire territory.

In this paper, the [Formula: see text]-Gordian complex of knots is defined. The authors show that for any knot [Formula: see text] and any positive integer [Formula: see text], there is a finite set of knots of size [Formula: see text] containing [Formula: see text], such that the [Formula: see text]-Gordian distance between any two knots in this set is 1 for all [Formula: see text].

The unknotting number of a welded knot is considered. First, we obtain an upper-bound of the unknotting number of a welded knot by using the warping degree method. Further, we introduce a standard welded torus knot with welded datum and obtain an upper bound of the unknotting number by an algorithm with warping degree method. Secondly, we get a lower bound of the unknotting number of a welded knot by Alexander quandle colorings. Finally, we give a definition of Gordian distance for welded knots analogous to the classical case.

We present formulae for computing the Yamada polynomial of spatial graphs obtained by replacing edges of plane graphs, such as cycle-graphs, theta-graphs, and bouquet-graphs, by spatial parts. As a corollary, it is shown that zeros of Yamada polynomials of some series of spatial graphs are dense in a certain region in the complex plane, described by a system of inequalities. Also, the relation between Yamada polynomial of graphs and the chain polynomial of edge-labelled graphs is obtained.

Let M i be a compact orientable 3-manifold, and A i a non-separating incompressible annulus on ∂ M i , i = 1, 2. Let h : A 1 → A 2 be a homeomorphism, and M = M 1 ∪ h M 2 the annulus sum of M 1 and M 2 along A 1 and A 2 . In the present paper, we show that if M i has a Heegaard splitting V i ∪ S i W i with distance d(S i ) 2g(M i ) + 3 for i = 1, 2, then g(M) = g(M 1 ) + g(M 2 ). Moreover, if g(F i ) 2, i = 1, 2, then the minimal Heegaard splitting of M is unique.