Topological nodal rings can be classified into three types according to the slopes in their energy dispersion. The first two are made of type-I and II nodal points, respectively, while the third is made of both. In carbon networks, all three types can exist. Under strain, phase transitions from a topological metal to a semiconductor, take place, and at the transition points, these nodal rings shrink into type-I, II, and III semi-Dirac points. These topological features exhibit diverse electron-hole pocket patterns and Landau levels, which give rise to exotic transport properties.Corresponding author: chenyp@xtu.edu.cn 2The study of topological semimetals/metals (TMs) are at the forefront of research in the materials and physical sciences [1][2][3][4]. With the discoveries of exotic band crossings near the Fermi level between two [5,6] or multiple bands [7][8][9], new topological concepts have emerged, for example, from the Dirac point [10,11] or Weyl point [12,13], to a nodal line [14][15][16], and from type-I to type-II topological points [17][18][19]. Each new finding expands the class of topological systems often with new physical properties. Therefore, searches for new classes of topological matter have been a recent focus [20,21].Adjusting band dispersion is an effective way to create new topological systems. The distinction between type-I and type-II Weyl points serves as a good example [22,23]. Both of these result from crossing points between linear bands having qualitatively different slopes. The conduction and valence bands of a type-II point have the same signs, while those of a type-I point have opposite signs [24,25]. Unlike relativistic high-energy physics, the violation of Lorentz invariance for type-II Weyl points reveals the unnecessary restriction of Lorentz symmetry in condensed matter systems [26]. Nevertheless type-I and II Weyl points share the same topological characteristics, and the two can be transformed from one to the other by a Lifshitz transition [27,28], and their Fermi surfaces, which are crucial to magnetic properties and carrier transport, can be drastically different [29,30].Because of its rich bonding chemistry, a carbon network provides a natural platform in the search for new topological matter [31][32][33], since carbon has many forms of allotropes ranging from one (1D) to three (3D)dimension [34][35][36][37][38][39][40]. Here, the presence of topological properties originates from graphene as a two-dimensional (2D) topological matter [41], with nanoribbons serving as building blocks for the carbon allotropes. The atomic structures of the carbon networks exhibit a wide variety of topological properties ranging from Weyl/Dirac points, nodal loops, to nodal surfaces [42,43]. 3D carbon honeycomb as one kind of carbon networks has been