Linkless and flat embeddings in 3-space and the unknot problem

• Authors: Ken-ichi Kawarabayashi;Stephan Kreutzer;Bojan Mohar
• Venue: Proceedings of the twenty-sixth annual symposium on Computational geometry
• Year: 2010

We consider piecewise linear embeddings of graphs in 3-space ℜ3. Such an embbeding is linkless if every pair of disjoint cycles forms a trivial link (in the sense of knot theory). Robertson, Seymour and Thomas [47] showed that a graph has a linkless embedding in ℜ3 if, and only if, it does not contain as a minor any of seven graphs in Petersen's family (graphs obtained from K6 by a series of YΔ and ΔY operations). They also showed that a graph is linklessly embeddable in ℜ3 if, and only if, it admits a flat embedding into ℜ3, i.e. an embedding such that for every cycle C of G there exists a closed 2-disk D ⊆ ℜ3 with D ∩ G = ∂D = C. Clearly, every flat embeddings is linkless, but the converse is not true. We first consider the following algorithmic problem associated with embeddings in ℜ3: Flat Embedding: For a given graph G, either detect one of Petersen's family graphs as a minor in G or return a flat (and hence linkless) embedding in ℜ3. The first outcome is a certificate that G has no linkless and no flat embeddings. Our first main result is to give an O(n2) algorithm for this problem. While there is a known polynomial-time algorithm for constructing linkless embeddings [20], this is the first polynomial time algorithm for constructing flat embeddings in 3-space and we thereby settle a problem proposed by Lovasz [29]. We also consider the following classical problem in topology. The Unknot Problem: Decide if a given knot is trivial or not. This is a fundamental problem in knot theory and low dimensional topology, whose time complexity is unresolved. It has been extensively studied by researchers working in computational geometry. A related problem is: The Link Problem: Decide if two given knots form a link. Hass, Lagarias and Pippenger [16] observed that a polynomial time algorithm for the link problem yields a polynomial time algorithm for the unknot problem. We relate the link problem to the following problem that was proposed independently by Lovasz and by Robertson et al. Conjecture. (Lovasz [29]; Robertson, Seymour and Thomas [48]) There is a polynomial time algorithm to decide whether a given embedding of a graph in the 3-space is linkless. Affirming this conjecture would clearly yield a polynomial-time solution for the link problem. We prove that the converse is also true by providing a polynomial-time solution for the above conjecture, if we are given a polynomial time oracle for the link problem.