Deligne-Beilinson cohomology and abelian link invariants: torsion case

For the abelian Chern-Simons field theory, we consider the quantum functional integration over the Deligne-Beilinson cohomology classes and present an explicit path-integral non-perturbative computation of the Chern-Simons link invariants in $SO(3)\simeq\mathbb{R}P^3$, a toy example of 3-manifold with torsion

Knots in interaction

We study the geometry of interacting knotted solitons. The interaction is local and advances either as a three-body or as a four-body process, depending on the relative orientation and a degeneracy of the solitons involved. The splitting and adjoining is governed by a four-point vertex in combination with duality transformations. The total linking number is preserved during the interaction. It receives contributions both from the twist and the writhe, which are variable. Therefore solitons can twine and coil and links can be formed.Comment: figures now in GIF forma

Surface-Invariants in 2D Classical Yang-Mills Theory

We study a method to obtain invariants under area-preserving diffeomorphisms associated to closed curves in the plane from classical Yang-Mills theory in two dimensions. Taking as starting point the Yang-Mills field coupled to non dynamical particles carrying chromo-electric charge, and by means of a perturbative scheme, we obtain the first two contributions to the on shell action, which are area-invariants. A geometrical interpretation of these invariants is given.Comment: 17 pages, 2 figure

Alexander quandle lower bounds for link genera

We denote by Q_F the family of the Alexander quandle structures supported by finite fields. For every k-component oriented link L, every partition P of L into h:=|P| sublinks, and every labelling z of such a partition by the natural numbers z_1,...,z_n, the number of X-colorings of any diagram of (L,z) is a well-defined invariant of (L,P), of the form q^(a_X(L,P,z)+1) for some natural number a_X(L,P,z). Letting X and z vary in Q_F and among the labellings of P, we define a derived invariant A_Q(L,P)=sup a_X(L,P,z). If P_M is such that |P_M|=k, we show that A_Q(L,P_M) is a lower bound for t(L), where t(L) is the tunnel number of L. If P is a "boundary partition" of L and g(L,P) denotes the infimum among the sums of the genera of a system of disjoint Seifert surfaces for the L_j's, then we show that A_Q(L,P) is at most 2g(L,P)+2k-|P|-1. We set A_Q(L):=A_Q(L,P_m), where |P_m|=1. By elaborating on a suitable version of a result by Inoue, we show that when L=K is a knot then A_Q(K) is bounded above by A(K), where A(K) is the breadth of the Alexander polynomial of K. However, for every g we exhibit examples of genus-g knots having the same Alexander polynomial but different quandle invariants A_Q. Moreover, in such examples A_Q provides sharp lower bounds for the genera of the knots. On the other hand, A_Q(L) can give better lower bounds on the genus than A(L), when L has at least two components. We show that in order to compute A_Q(L) it is enough to consider only colorings with respect to the constant labelling z=1. In the case when L=K is a knot, if either A_Q(K)=A(K) or A_Q(K) provides a sharp lower bound for the knot genus, or if A_Q(K)=1, then A_Q(K) can be realized by means of the proper subfamily of quandles X=(F_p,*), where p varies among the odd prime numbers.Comment: 36 pages; 16 figure

A Theorem of Sanderson on Link Bordisms in Dimension 4

The groups of link bordism can be identified with homotopy groups via the Pontryagin-Thom construction. B.J. Sanderson computed the bordism group of 3 component surface-links using the Hilton-Milnor Theorem, and later gave a geometric interpretation of the groups in terms of intersections of Seifert hypersurfaces and their framings. In this paper, we geometrically represent every element of the bordism group uniquely by a certain standard form of a surface-link, a generalization of a Hopf link. The standard forms give rise to an inverse of Sanderson's geometrically defined invariant.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol1/agt-1-14.abs.htm

Abelian BF theory and Turaev-Viro invariant

The U(1) BF Quantum Field Theory is revisited in the light of Deligne-Beilinson Cohomology. We show how the U(1) Chern-Simons partition function is related to the BF one and how the latter on its turn coincides with an abelian Turaev-Viro invariant. Significant differences compared to the non-abelian case are highlighted.Comment: 47 pages and 6 figure

Lens Spaces and Handlebodies in 3D Quantum Gravity

We calculate partition functions for lens spaces L_{p,q} up to p=8 and for genus 1 and 2 handlebodies H_1, H_2 in the Turaev-Viro framework. These can be interpreted as transition amplitudes in 3D quantum gravity. In the case of lens spaces L_{p,q} these are vacuum-to-vacuum amplitudes \O -> \O, whereas for the 1- and 2-handlebodies H_1, H_2 they represent genuinely topological transition amplitudes \O -> T^2 and \O -> T^2 # T^2, respectively.Comment: 14 pages, LaTeX, 5 figures, uses eps

On the Kauffman bracket skein module of the quaternionic manifold

We use recoupling theory to study the Kauffman bracket skein module of the quaternionic manifold over Z[A,A^{-1}] localized by inverting all the cyclotomic polynomials. We prove that the skein module is spanned by five elements. Using the quantum invariants of these skein elements and the Z_2 homology of the manifold, we determine that they are linearly independent.Comment: corrected summation signs in figures 14, 15, 17. Other minor change

Supersymmetry, homology with twisted coefficients and n-dimensional knots

Let $n$ be any natural number. Let $K$ be any $n$-dimensional knot in $S^{n+2}$. We define a supersymmetric quantum system for $K$ with the following properties. We firstly construct a set of functional spaces (spaces of fermionic \{resp. bosonic\} states) and a set of operators (supersymmetric infinitesimal transformations) in an explicit way. Thus we obtain a set of the Witten indexes for $K$. Our Witten indexes are topological invariants for $n$-dimensional knots. Our Witten indexes are not zero in general. If $K$ is equivalent to the trivial knot, all of our Witten indexes are zero. Our Witten indexes restrict the Alexander polynomials of $n$-knots. If one of our Witten indexes for an $n$-knot $K$ is nonzero, then one of the Alexander polynomials of $K$ is nontrivial. Our Witten indexes are connected with homology with twisted coefficients. Roughly speaking, our Witten indexes have path integral representation by using a usual manner of supersymmetric theory.Comment: 10pages, no figure

Topological Aspect of Knotted Vortex Filaments in Excitable Media

Scroll waves exist ubiquitously in three-dimensional excitable media. It's rotation center can be regarded as a topological object called vortex filament. In three-dimensional space, the vortex filaments usually form closed loops, and even linked and knotted. In this letter, we give a rigorous topological description of knotted vortex filaments. By using the $\phi$-mapping topological current theory, we rewrite the topological current form of the charge density of vortex filaments and use this topological current we reveal that the Hopf invariant of vortex filaments is just the sum of the linking and self-linking numbers of the knotted vortex filaments. We think that the precise expression of the Hopf invariant may imply a new topological constraint on knotted vortex filaments.Comment: 4 pages, no figures, Accepted by Chin. Phys. Let