Cobordism, homotopy and homology of graphs in R3

Vassiliev invariants of knots in a spatial graph

“…We note that edge-homotopy and vertex-homotopy are equivalence relations on spatial graphs introduced in [18] as generalizations of Milnor's link homotopy [9]. We remark that edge-homotopy implies vertex-homotopy since G is loopless [18].…”

“…We also note that an edge-homotopy invariant for D 3 defined in [18] is an example of Theorem 3.4 where k = n = 2, R = Z/2Z and v γ is the Arf invariant of knots. In general it follows from Theorem 3.4 that the spatial graphs in Figure 1.2 are not edge-homotopic to plane graphs.…”

An unknotting theorem for delta and sharp edge‐homotopy

“…Since each of K 5 and K 3,3 contains a subgraph which is homeomorphic to Θ, we have that G is a planar graph which does not contain mutually disjoint cycles. Thus by [23,Theorem C] we have that G is homeomorphic to a double trident, a multi-spoke wheel or a generalized bouquet. Here a double trident and a multi-spoke wheel are graphs as illustrated in Fig.…”

“…Therefore this equivalence relation does not depend on the edge orientations. Edge-homotopy on spatial graphs was introduced by Taniyama in [23] as a generalization of link homotopy in the sense of Milnor [8]. Delta edgehomotopy and sharp edge-homotopy on spatial graphs were introduced by the author in [15] and [17] as generalizations of self ∆-equivalence [22] (or delta link homotopy [12]) and self ♯-equivalence [20] on oriented links, respectively.…”

An intrinsic nontriviality of graphs

“…Here, a Delta equivalence is an equivalence relation on spatial graphs which is generated by Delta moves and ambient isotopies, where a Delta move is a local move on a spatial graph as illustrated in Figure 8 [8; 10]. It is shown in [9] that a Delta equivalence Figure 8 coincides with a (spatial graph-)homology, which is an equivalence relation on spatial graphs introduced in [18]. Note that a Delta move preserves the linking number of each of the constituent 2-component links.…”