Journal of Knot Theory and Its Ramifications

Intersection homology enables the definition of a twisted signature of a stratified pseudomanifold with coefficients in a local system that is typically only given on the top stratum. If the local system extends to the entire space, then the twisted signature can be computed by a version of a characteristic class formula first observed by Atiyah for smooth fiber bundles. In the first part of this paper, we construct examples of singular spaces, equipped with local systems that do not extend, such that the above characteristic class formula fails. In the second part, we consider smooth codimension two embeddings of manifolds. We view the target as a stratified space with bottom stratum the image of the embedding and top stratum the complement. When the embedded manifold is a sphere, we establish various formulae that compute the twisted signature even when the local system does not extend from the top stratum to the entire space.

The normal holonomy of a smooth knot in Euclidean space is a geometric invariant with value in the unit circle which is closely related to the writhing number. Here we prove that normal holonomy fibers the space of smooth knots, except for possible singularities on the set of round knots. We deduce results on the existence of isotopies of constant holonomy, writhe and twist.

We introduce the notions of "k-connected-slice" and "π₁-slice", interpolating between "homotopy ribbon" and "slice". Every high-dimensional knot group π is the group of an (n - 1)-connected-slice n-knot, for all n ≥ 3. However, if π is the group of an n-connected-slice n-knot, the augmentation ideal I(π) has deficiency 1 as a module, while (n + 1)-connected-slice n-knots are trivial. If π is the group of a π₁-slice 2-knot and π' is finitely generated, then π' is free, and so def(π) = 1.

We exhibit an infinite family of knots that are detected chiral by the Kauffman polynomial but not by the HOMFLY polynomial.

We show that no torus knot of type (2, n), n > 3 odd, can be obtained from a polynomial embedding t ↦ (f(t), g(t), h(t)) where (deg(f), deg(g)) ≤ (3, n + 1). Eventually, we give explicit examples with minimal lexicographic degree.

A knot diagram can be divided by a circle into two parts, such that each part can be coded by a planar tree with integer weights on its edges. A half of the number of intersection points of this circle with the knot diagram is called the girth. The girth of a knot is the minimal girth of all diagrams of this knot. The girth of a knot minus one is an upper bound of the Heegaard genus of the 2-fold branched covering of that knot. We will use Topological Quantum Field Theory (TQFT) coming from the Kauffman bracket to determine the girth of some knots. Consequently, our method can be used to determine the Heegaard genus of the 2-fold branched covering of some knots.

In analogy with a recursive formula for the HOMFLY-PT polynomial of links given by Jaeger, we give a recursive formula for the graph polynomial introduced by Kauffman and Vogel. We show how this formula extends to the Khovanov–Rozansky graph homology.

We show that the Khovanov complex of a connected link diagram D retracts to a subcomplex whose generators are in 2:1 correspondence with the spanning trees of the "black graph" of D. Using this result, we give a new proof of Lee's theorem on the support of Khovanov homology of alternating knots.

Given an edge-oriented polygonal graph in ℝ³, we describe a method for computing the writhe as the average of weighted directional writhe numbers of the graph in a few directions. These directions are determined by the graph and the weights are determined by areas of path-connected open regions on the unit sphere. Within each open region, the directional writhe is constant. We obtain a closed formula which extends the formula for the writhe of a polygon in ℝ³, including the important special case of writhe of embedded open arcs.