Dijkgraaf Witten Invariants Research Articles

Alexander polynomial, Dijkgraaf-Witten invariant, and Seifert fibred surgery

We apply Dijkgraaf-Witten invariants over semidirect products of abelian groups to show that, if the k/ℓ-surgery along a knot K results in a small Seifert 3-manifold with multiplicities a1,a2,a3, then many constraints on k,a1,a2,a3 can be read off from the Alexander polynomial of K.

The reduced Dijkgraaf–Witten invariant of double twist knots in the Bloch group of 𝔽p

In 2004, Neumann showed that the complex hyperbolic volume of a hyperbolic 3-manifold [Formula: see text] can be obtained as the image of the Dijkgraaf–Witten invariant of [Formula: see text] by a certain 3-cocycle. After that, Zickert gave an analogue of Neumann’s work for free fields containing finite fields. The author formulated a geometric method to calculate a weaker version of Zickert’s analogue, called the reduced Dijkgraaf–Witten invariant, for finite fields and gave a formula for twist knot complements and [Formula: see text] in his previous work. In this paper, we show concretely how to calculate the reduced Dijkgraaf–Witten invariants of double twist knot complements and [Formula: see text], and give a formula of them for [Formula: see text].

The reduced Dijkgraaf–Witten invariant of twist knots in the Bloch group of a finite field

Let [Formula: see text] be a closed oriented 3-manifold and let [Formula: see text] be a discrete group. We consider a representation [Formula: see text]. For a 3-cocycle [Formula: see text], the Dijkgraaf–Witten invariant is given by [Formula: see text], where [Formula: see text] is the map induced by [Formula: see text], and [Formula: see text] denotes the fundamental class of [Formula: see text]. Note that [Formula: see text], where [Formula: see text] is the map induced by [Formula: see text], we consider an equivalent invariant [Formula: see text], and we also regard it as the Dijkgraaf–Witten invariant. In 2004, Neumann described the complex hyperbolic volume of [Formula: see text] in terms of the image of the Dijkgraaf–Witten invariant for [Formula: see text] by the Bloch–Wigner map from [Formula: see text] to the Bloch group of [Formula: see text]. In this paper, by replacing [Formula: see text] with a finite field [Formula: see text], we calculate the reduced Dijkgraaf–Witten invariants of the complements of twist knots, where the reduced Dijkgraaf–Witten invariant is the image of the Dijkgraaf–Witten invariant for SL[Formula: see text] by the Bloch–Wigner map from [Formula: see text] to the Bloch group of [Formula: see text].

On the monoidal center of Deligne's category Re̲p(St)

We explicitly compute a monoidal subcategory of the monoidal center of Deligne's interpolation category Rep(S_t), for t not necessarily a natural number, and we show that this subcategory is a ribbon category. For t=n, a natural number, there exists a functor onto the braided monoidal category of modules over the Drinfeld double of S_n which is essentially surjective and full. Hence the new ribbon category interpolates the categories of crossed modules over the symmetric groups. As an application, we obtain invariants of framed ribbon links which are polynomials in the interpolating variable t. These polynomials interpolate untwisted Dijkgraaf-Witten invariants of the symmetric groups.

EXPLICIT cocycle formulas on finite abelian groups with applications to braided linear Gr-categories and Dijkgraaf–Witten invariants

AbstractWe provide explicit and unified formulas for the cocycles of all degrees on the normalized bar resolutions of finite abelian groups. This is achieved by constructing a chain map from the normalized bar resolution to a Koszul-like resolution for any given finite abelian group. With a help of the obtained cocycle formulas, we determine all the braided linear Gr-categories and compute the Dijkgraaf–Witten Invariants of the n-torus for all n.

The Dijkgraaf–Witten invariants of Seifert 3-manifolds with orientable bases

We derive a formula for the Dijkgraaf–Witten invariants of Seifert 3-manifolds with orientable bases.

On third homologies of groups and of quandles via the Dijkgraaf–Witten invariant and Inoue–Kabaya map

We propose a simple method to produce quandle cocycles from group cocycles, as a modification of Inoue-Kabaya chain map. We further show that, in respect to "universal central extended quandles", the chain map induces an isomorphism between their third homologies. For example, all Mochizuki's quandle 3-cocycles are shown to be derived from group cocycles of some non-abelian group. As an application, we calculate some $\Z$-equivariant parts of the Dijkgraaf-Witten invariants of some cyclic branched covering spaces, via some cocycle invariant of links.

Counting homotopy classes of mappings via Dijkgraaf–Witten invariants

Suppose Γ is a finite group acting freely on Sn (n≥3 being odd) and M is any closed oriented n-manifold. We show that, given an integer k, the set deg−1(k) of based homotopy classes of mappings with degree k is finite and its cardinality depends only on the congruence class of k modulo #Γ; moreover, #deg−1(k) can be expressed in terms of the Dijkgraaf–Witten invariants of M.

SOME TOPOLOGICAL ASPECTS OF 4-FOLD SYMMETRIC QUANDLE INVARIANTS OF 3-MANIFOLDS

The paper relates the 4-fold symmetric quandle homotopy (cocycle) invariants to topological objects. We show that the 4-fold symmetric quandle homotopy invariants are at least as powerful as the Dijkgraaf–Witten invariants. As an application, for an odd prime p, we show that the quandle cocycle invariant of a link in S3 constructed by the Mochizuki 3-cocycle is equivalent to the Dijkgraaf–Witten invariant with respect to ℤ/pℤ of the double covering of S3 branched along the link. We also reconstruct the Chern–Simons invariant of closed 3-manifolds as a quandle cocycle invariant via the extended Bloch group, in analogy to [A. Inoue and Y. Kabaya, Quandle homology and complex volume, preprint(2010), arXiv:math/1012.2923].

ON TURAEV'S THEOREM ABOUT DIJKGRAAF–WITTEN INVARIANTS OF SURFACES

Turaev proves a formula for the Dijkgraaf–Witten invariants of surfaces in terms of projective representations by using the state sum invariant technique from quantum topology. In this paper, we present another proof of Turaev's theorem by using classical method of characters and representation theory. A version of Turaev's formula for surfaces with boundary is also given.

Dijkgraaf–Witten invariants of surfaces and projective representations of groups

We compute the Dijkgraaf–Witten invariants of surfaces in terms of projective representations of groups. As an application we prove that the complex Dijkgraaf–Witten invariants of surfaces of positive genus are positive integers.

ON THE TOPOLOGICAL ORIGIN OF ENTANGLEMENT IN ISING SPIN GLASSES

The origin of entanglement in a class of three-dimensional spin models, at low momenta, is traced to topological reasons. The establishment of the result is facilitated by the gauge principle which, in conjunction with the duality mapping of the spin models, enables us to recast them as lattice Chern–Simons theories. The entanglement measures are expressed in terms of the correlators of Wilson lines, loops, and their generalisations. For continuous spins, these yield the invariants of knots and links. For Ising-like models, they can be expressed in terms of three-manifold invariants obtained from finite group cohomology — the so-called Dijkgraaf–Witten invariants.

Two Physicists Claim Share of Dutch Science Prize

In February, the Netherlands Organisation for Scientific Research (NWO) will honor the four recipients of the 2003 NWO/Spinoza Prize, the most prestigious Dutch award for science. During the ceremony in The Hague, each of the awardees will receive both a statuette and ¢1.5 million (about $1.7 million) to use for research. Cees Dekker, professor of molecular biophysics at the Delft University of Technology in the Netherlands, is being recognized in particular for his work on carbon nanotubes. According to the NWO, Dekker and his group “were the first in the world to measure the electrical conductivity of a single molecule … [and] developed the first nanotransistor, which was based on a nanotube.” His most recent work is in the emerging field of nanobiophysics. Robbert Dijkgraaf will receive the prize for his contributions to mathematics as well as for his work on string theory. His contributions include mathematical concepts important to physics, such as the Dijkgraaf-Witten invariants and the Witten-Dijkgraaf-Verlinde-Verlinde equations. Dijkgraaf was a doctoral student of Gerard’t Hooft, a 1995 winner of the NWO/Spinoza Prize; that makes Dijkgraaf the first Spinoza laureate to have studied under a Spinoza winner. He is a professor of mathematical physics at the University of Amsterdam.© 2003 American Institute of Physics.

The cohomology ring of the orientable Seifert manifolds, II

The cohomology ring of an arbitrary orientable Seifert manifold is computed with Z/p coefficients for any prime p. In some cases the cohomology rings are given with Z/p s coefficients. These results will be used to compute the abelian Witten–Reshetikhin–Turaev type invariants and Dijkgraaf–Witten invariants for some classes of Seifert manifolds in a later paper. Finally, necessary and sufficient conditions for the existence of a degree one map from an orientable Seifert manifold into a lens space are given.