Invariants of Boundary Link Cobordism

The Levine-Tristram signature associates to each oriented link $L$ in $S^3$ a function $\sigma_L \colon S^1 \to \mathbb{Z}.$ This invariant can be defined in a variety of ways, and its numerous applications include the study of unlinking numbers and link concordance. In this survey, we recall the three and four dimensional definitions of $\sigma_L$, list its main properties and applications, and give comprehensive references for the proofs of these statements.

We consider a class of topological objects in the 3-sphere $S^3$ which will be called $n$-punctured ball tangles. Using the Kauffman bracket at $A=e^{i \pi/4}$, an invariant for a special type of $n$-punctured ball tangles is defined. The invariant $F^n$ takes values in $PM_{2\times2^n}(\mathbb Z)$, that is the set of $2\times 2^n$ matrices over $\mathbb Z$ modulo the scalar multiplication of $\pm1$. This invariant leads to a generalization of a theorem of D. Krebes which giv...

Let N_1, N_2, M be smooth manifolds with dim N_1 + dim N_2 +1 = dim M$ and let phi_i, for i=1,2, be smooth mappings of N_i to M with Im phi_1 and Im phi_2 disjoint. The classical linking number lk(phi_1,phi_2) is defined only when phi_1*[N_1] = phi_2*[N_2] = 0 in H_*(M). The affine linking invariant alk is a generalization of lk to the case where phi_1*[N_1] or phi_2*[N_2] are not zero-homologous. In arXiv:math.GT/0207219 we constructed the first examples of affine linking ...

In the present paper, we construct a simple invariant which provides a sliceness obstruction for {\em free knots}. This obstruction provides a new point of view to the problem of studying cobordisms of curves immersed in 2-surfaces, a problem previously studied by Carter, Turaev, Orr, and others. The obstruction to sliceness is constructed by using the notion of {\em parity} recently introduced by the author into the study of virtual knots and their modifications. This inva...

We introduce a Heegaard-Floer homology functor from the category of oriented links in closed $3$-manifolds and oriented surface cobordisms in $4$-manifolds connecting them to the category of $\mathbb{F}[v]$-modules and $\mathbb{F}[v]$-homomorphisms between them, where $\mathbb{F}$ is the field with two elements. In comparison with previously defined TQFTs for decorated links and link cobordisms, the construction of this paper has the advantage of being independent from the de...

A new topological invariant of closed connected orientable four-dimensional manifolds is proposed. The invariant, constructed via surgery on a special link, is a four-dimensional counterpart of the celebrated SU(2) three-manifold invariant of Reshetikhin, Turaev and Witten.

We classify compact oriented $5$-manifolds with free fundamental group and $\pi_{2}$ a torsion free abelian group in terms of the second homotopy group considered as $\pi_1$-module, the cup product on the second cohomology of the universal covering, and the second Stiefel-Whitney class of the universal covering. We apply this to the classification of simple boundary links of $3$-spheres in $S^5$. Using this we give a complete algebraic picture of closed $5$-manifolds with fre...

We give an $O(p^{2})$ time algorithm to compute the generalized Heegaard Floer complexes $A_{s_{1},s_{2}}^{-}(\overrightarrow{L})$'s for a two-bridge link $\overrightarrow{L}=b(p,q)$ by using nice diagrams. Using the link surgery formula of Manolescu-Ozsv\'{a}th, we also show that ${\bf HF}^{-}$ and their $d$-invariants of all integer surgeries on two-bridge links are determined by $A_{s_{1},s_{2}}^{-}(\overrightarrow{L})$'s. We obtain a polynomial time algorithm to compute $...

This paper describes how to compute algorithmically certain twisted signature invariants of a knot $K$ using twisted Blanchfield forms. An illustration of the algorithm is implemented on $(2,q)$-torus knots. Additionally, using satellite formulas for these invariants, we also show how to obstruct the sliceness of certain iterated torus knots.

We study framed links in irreducible 3-manifolds that are $Z$-homology 3-spheres or atoroidal $Q$-homology 3-spheres. We calculate the dual of the Kauffman skein module over the ring of two variable power series with complex coefficients. For links in $S^3$ we give a new construction of the classical Kauffman polynomial.