Journal of Knot Theory and Its Ramifications

We describe a relationship between the crossing number of a graph G with a 2-edge-cut C and the crossing numbers of the components of G-C. Let G be a connected graph with a 2-edge-cut C := [V₁,V₂]. Let u₁u₂, v₁v₂ be the edges of C, so that u_i,v_i ∈ V_i for i = 1,2, and let G_i := G[V_i] and G'_i := G_i + u_iv_i. We show that if either G₁ or G₂ is not connected, then cr(G) = cr(G₁) + cr(G₂), and that if they are both connected then cr(G) = cr(G'₁) + cr(G'₂). We use this to show how to decompose crossing-critical graphs with 2-edge-cuts into smaller, 3-edge-connected crossing-critical graphs. We also observe that this settles a question arising from knot theory, raised by Sawollek, by describing exactly under which conditions the crossing number of the connected sum of two graphs equals the sum of the crossing numbers of the individual graphs.

It is conjectured that a hyperbolic knot admits at most three Dehn surgeries which yield closed 3-manifolds containing incompressible tori. We show that there exist infinitely many hyperbolic knots which attain the conjectural maximum number. Interestingly, those surgeries correspond to consecutive integers.

Previously, we constructed an infinite family of knotted symplectic tori representing a fixed homology class in the symplectic four-manifold E(n)_K, which is obtained by Fintushel–Stern knot surgery using a nontrivial fibered knot K in S³, and distinguished the (smooth) isotopy classes of these tori by indirectly computing the Seiberg–Witten invariants of their complements. In this note, we compute the fundamental groups of the complements of these knotted tori and show that for each nontrivial fibered knot K these groups constitute an infinite collection of nonisomorphic groups. We also review some other constructions of symplectic tori in 4-manifolds and show that the fundamental groups of the complements do not distinguish homologous tori in those cases.

By obtaining surgery descriptions of knots which lie on the genus one fiber of the trefoil or figure eight knot, we show that these include hyperbolic knots with arbitrarily large volume. These knots admit lens space surgeries and form two families of Berge knots. By way of tangle descriptions we also obtain surgery descriptions for these knots on minimally twisted chain links.

Using Kirby calculus, we explicitly pass from Berge's R-R descriptions of ten families of knots with lens space surgeries to surgery descriptions on the minimally twisted five chain link (MT5C). Since the MT5C admits a strong inversion, we also give the corresponding tangle descriptions.

We reconsider the link homology theory defined by Knovanov in [9] and generalized by Mackaay and Vaz in [15]. With some slight modifications, we describe the theory as a map from the planar algebra of tangles to a planar algebra of complexes of "cobordisms with seams" (actually, a "canopolis"), making it local in the sense of Bar-Natan's local theory of [2].

We show that this "seamed cobordism canopolis" decategorifies to give precisely what you had both hope for and expect: Kuperberg's spider defined in [14]. We conjecture an answer to an even more interesting question about the decategorification of the Karoubi envelope of our cobordism theory.

Finally, we describe how the theory is actually completely computable, and give a detailed calculation of the homology of the (2,n) torus knots.