Journal of Knot Theory and Its Ramifications

It is well-known that the "all right" metric structure on the Dehn complex of a knot is non-positively curved, or CAT(0), if and only if the knot projection is prime and alternating. We extend these ideas to spatial graphs and introduce the idea of ψ-alternating graphs as the appropriate analog of alternating knots.

We construct a series of combinatorial quandle-like knot invariants. We color regions of a knot diagram rather than lines and assign a weight to each coloring. Sets of these weights are the invariants we construct (colorings and weights depend on several parameters).

Using these invariants, we prove that left and right trefoils are not isotopic using this invariant (in a particular case).

We show that if a genus two hyperbolic knot K in S³ has a toroidal slope r, then |r| ≤ 8. This settles a conjecture of Teragaito for a special case.

In this paper, we introduce the notion of Reidemeister torsion for quasi-isomorphisms of based chain complexes over a field. We call a chain map a quasi-isomorphism if its induced homomorphism between homology is an isomorphism. Our notion of torsion generalizes the torsion of acyclic based chain complexes, and is a chain homotopy invariant on the collection of all quasi-isomorphisms from a based chain complex to another. It shares nice properties with torsion of acyclic based chain complexes, like multiplicativity and duality. We will further generalize our torsion to quasi-isomorphisms between free chain complexes over a ring under some mild condition. We anticipate that the study of torsion of quasi-isomorphisms will be fruitful in many directions, and in particular, in the study of links in 3-manifolds.

Colorings of torus knots and twist-spun torus knots by general Alexander quandles over finite fields are explicitly determined. It is then observed that, for some standard Mochizuki's cocycles, a large class of twist-spun torus knots with non-trivial colorings have trivial CJKLS state-sum invariants.

We calculate certain limits of the colored Alexander invariant of the figure-eight knot for some cases and show that this limit is related to the volume of hyperbolic orbifolds whose singular set is the figure-eight knot.

Satoh has defined a construction to obtain a ribbon torus knot given a welded knot. This construction is known to be surjective. We show that it is not injective. Using the invariant of the peripheral structure, it is possible to provide a restriction on this failure of injectivity. In particular we also provide an algebraic classification of the construction when restricted to classical knots, where it is equivalent to the torus spinning construction.

The ℂP²-genus of a knot K is the minimal genus over all isotopy classes of smooth, compact, connected and oriented surfaces properly embedded in ℂP² - B⁴ with boundary K. We compute the ℂP²-genus and realizable degrees of (-2,q)-torus knots for 3 ≤ q ≤ 11 and (2,q)-torus knots for 3 ≤ q ≤ 17. The proofs use gauge theory and twisting operations on knots.

We show how to use Bar-Natan's "divide and conquer" approach to computation to efficiently compute the universal sl(2) dotted foam cohomology groups, even for big knots and links. We also describe a purely topological version of the sl(2) foam theory, in the sense that no dots are needed on foams.