Journal of Knot Theory and Its Ramifications

We give several new positive finite presentations for the pure braid group that are easy to remember and simple in form. All of our presentations involve a metric on the punctured disc so that the punctures are arranged "convexly", which is why we describe them as geometric presentations. Motivated by a presentation for the full braid group that we call the "rotation presentation", we introduce presentations for the pure braid group that we call the "twist presentation" and the "swing presentation". From the point of view of mapping class groups, the swing presentation can be interpreted as stating that the pure braid group is generated by a finite number of Dehn twists and that the only relations needed are the disjointness relation and the lantern relation.

We give surgery formulas for the Ohtsuki invariants λ₁, λ₂ and λ₃ of an integral homology sphere obtained by surgery from S³ along a knot, related to Habiro's cyclotomic expansion of the colored Jones polynomial of the knot. As an application, we prove that the Ohtsuki invariants λ₁, λ₂ and λ₃ separate integral homology 3-spheres obtained from S³ by surgery along the Borromean rings.

For a closed oriented 3-manifold M and an integer r > 0, let τ_r(M) denote the SU(2) Reshetikhin–Turaev–Witten invariant of M at level r. We show that for every n > 0, and for r₁,…, r_n > 0 sufficiently large integers, there exist infinitely many non-homeomorphic hyperbolic 3-manifolds M, all of which have different hyperbolic volume, such that τ_ri(M) = 1, for i = 1,…, n.

We often study surface links in 4-space by using their projections into 3-space, or their broken surface diagrams. It is well-known that a broken surface diagram recovers the given surface link. In this paper, we study surface links in 4-space by using their generic projections into the plane. These projections have fold points and cusps as their singularities in general. We consider the question whether such a generic planar projection can recover the given surface link. We introduce the notion of banded braids, and show that a generic planar projection together with banded braids associated to the segments of the fold curve image can recover the given surface link. As an application, we give a new proof to the Whitney congruence concerning the normal Euler number of surface links.

We show that one can interweave an unknot into any non-alternating connected projection of a link so that the resulting augmented projection is alternating. As a corollary, we strengthen a result of Myers in the case of link complements in S³.

We show that the commutator subgroup G′ of a classical knot group G need not have subgroups of every finite index, but it will if G′ has a surjective homomorphism to the integers and we give an exact criterion for that to happen. We also give an example of a knotted Sⁿ in Sⁿ⁺² for all n ≥ 2 whose infinite cyclic cover is not simply connected but has no proper finite covers.

We use a special kind of 2-dimensional extended Topological Quantum Field Theories (TQFTs), so-called open-closed TQFTs, in order to represent a refinement of Bar-Natan's universal geometric complex algebraically, and thereby extend Khovanov homology from links to arbitrary tangles. For every plane diagram of an oriented tangle, we construct a chain complex whose terms are modules of a suitable algebra A such that there is one action of A or A^op for every boundary point of the tangle. We give examples of such algebras A for which our tangle homology theory reduces to the link homology theories of Khovanov, Lee and Bar-Natan if it is evaluated for links. As a consequence of the Cardy condition, Khovanov's graded theory can only be extended to tangles if the underlying field has finite characteristic. Whenever the open-closed TQFT arises from a state-sum construction, we obtain honest planar algebra morphisms, and all composition properties of the universal geometric complex carry over to the algebraic complex. We give examples of state-sum open-closed TQFTs for which one can still determine both characteristic p Khovanov homology of links and Rasmussen's s-invariant.