Narrow Results

We study an exact asymptotic behavior of the Witten–Reshetikhin–Turaev SU(2) invariant for the Brieskorn homology spheres Σ(p₁, p₂, p₃) by use of properties of the modular form following a method proposed by Lawrence and Zagier. Key observation is that the invariant coincides with a limiting value of the Eichler integral of the modular form with weight 3/2. We show that the Casson invariant is related to the number of the Eichler integrals which do not vanish in a limit τ → N ∈ ℤ. Correspondingly there is a one-to-one correspondence between the non-vanishing Eichler integrals and the irreducible representation of the fundamental group, and the Chern–Simons invariant is given from the Eichler integral in this limit. It is also shown that the Ohtsuki invariant follows from a nearly modular property of the Eichler integral, and we give an explicit form in terms of the L-function.

It is known that the LMO invariant of 3-manifolds with positive first Betti numbers is relatively weak and can be determined by “(semi-)classical” invariants such as the cohomology ring, the Alexander polynomial, and the Casson–Walker–Lescop invariant.

In this paper, we formulate a refinement of the LMO invariant for 3-manifolds with the first Betti number 1. It dominates the perturbative SO(3) invariant of such 3-manifolds, which is the power series invariant formulated by the arithmetic perturbative expansion of the quantum SO(3) invariants of such 3-manifolds. As the 2-loop part of the refinement of the LMO invariant, we define the 2-loop polynomial of such 3-manifolds. Further, as the $𝔰{𝔩}_m$ reduction at large $m$ limit of the $\ell$-loop part of the refinement of the LMO invariant for $\ell \le 5$, we formulate an $\ell$-variable polynomial invariant of such 3-manifolds whose Alexander polynomial is constant.

We use recoupling theory to study the Kauffman bracket skein module of the quaternionic manifold over ℤ[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 ℤ₂-homology of the manifold, we determine that they are linearly independent.

This paper gives a new interpretation of the virtual braid group in terms of a strict monoidal category SC that is freely generated by one object and three morphisms, two of the morphisms corresponding to basic pure virtual braids and one morphism corresponding to a transposition in the symmetric group. The key to this approach is to take pure virtual braids as primary. The generators of the pure virtual braid group are abstract solutions to the algebraic Yang–Baxter equation. This point of view illuminates representations of the virtual braid groups and pure virtual braid groups via solutions to the algebraic Yang–Baxter equation. In this categorical framework, the virtual braid group is a natural group associated with the structure of algebraic braiding. We then point out how the category SC is related to categories associated with quantum algebras and Hopf algebras and with quantum invariants of virtual links.

We show that for any n ≥ 4 there exists an equivalence functor from the category of n-fold connected simple coverings of B³ × [0, 1] branched over ribbon surface tangles up to certain local ribbon moves, and the cobordism category of orientable relative 4-dimensional 2-handlebody cobordisms up to 2-deformations. As a consequence, we obtain an equivalence theorem for simple coverings of S³ branched over links, which provides a complete solution to the long-standing Fox–Montesinos covering moves problem. This last result generalizes to coverings of any degree results by the second author and Apostolakis, concerning respectively the case of degree 3 and 4. We also provide an extension of the equivalence theorem to possibly non-simple coverings of S³ branched over embedded graphs. Then, we factor the functor above as , where is an equivalence functor to a universal braided category freely generated by a Hopf algebra object H. In this way, we get a complete algebraic description of the category . From this we derive an analogous description of the category of 2-framed relative 3-dimensional cobordisms, which resolves a problem posed by Kerler.

We construct quantum type invariants for handlebody-knots in the 3-sphere S³. A handlebody-knot is an embedding of a handlebody in a 3-manifold. These invariants are linear sums of Yokota's invariants for colored spatial graphs which are defined by using the Kauffman bracket. We give a table of calculations of our invariants for genus 2 handlebody-knots up to six crossings. We also show our invariants are identified with special cases of the Witten–Reshetikhin–Turaev invariants.

To systematically construct invariants of handlebody-links, we give a new presentation of the braided tensor category of handlebody-tangles by generators and relations, and prove that given what we call a quantum-commutative quantum-symmetric algebra A in an arbitrary braided tensor category , there arises a braided tensor functor , which gives rise to a desired invariant. Some properties of the invariants and explicit computational results are shown especially when A is a finite-dimensional unimodular Hopf algebra, which is naturally regarded as a quantum-commutative quantum-symmetric algebra in the braided tensor category of Yetter–Drinfeld modules.

This paper studies rotational virtual knot theory and its relationship with quantum link invariants. Every quantum link invariant for classical knots and links extends to an invariant of rotational virtual knots and links. We give examples of non-trivial rotational virtuals that are undectable by quantum invariants.

We construct graph-valued analogues of the Kuperberg sl(3) and G₂ invariants for virtual knots. The restriction of the sl(3) and G₂ invariants for classical knots coincides with the usual Homflypt sl(3) invariant and G₂ invariants. For virtual knots and graphs these invariants provide new graphical information that allows one to prove minimality theorems and to construct new invariants for free knots (unoriented and unlabeled Gauss codes taken up to abstract Reidemeister moves). A novel feature of this approach is that some knots are of sufficient complexity that they evaluate themselves in the sense that the invariant is the knot itself seen as a combinatorial structure. The paper generalizes these structures to virtual braids and discusses the relationship with the original Penrose bracket for graph colorings.

In this study, we introduce new criteria for a given link to detect the non-$p$-periodicity for a prime $p\ge 5$. For this we use a congruence of the quantum ${𝔰𝔩}_N$-invariant of periodic links. Our result is naturally consistent with Traczyk’s criterion and Przytycki’s criterion, and can be regarded as a generalization of these classical criteria. We shall give computational results of all the (alternating and non-alternating) knots of crossing number $\le 16$ for their non-$p$-periodicity ($p\ge 5$). The computational results also show that our criteria has somewhat different nature with the classical criterion of Murasugi.

We consider the problem of distinguishing mutant knots using invariants of their satellites. We show, by explicit calculation, that the Homfly polynomial of the 3-parallel (and hence the related quantum invariants) will distinguish some mutant pairs.

Having established a condition on the colouring module which forces a quantum invariant to agree on mutants, we explain several features of the difference between the Homfly polynomials of satellites constructed from mutants using more general patterns. We illustrate this by our calculations; from these we isolate some simple quantum invariants, and a framed Vassiliev invariant of type 11, which distinguish certain mutants, including the Conway and Kinoshita-Teresaka pair.

For Z/rZ-homology 3-spheres the Casson invariant is related to the quantum PSU(2) invariant at an rth root of unity with odd prime r. We prove a similar relationship between the Casson invariant and the quantum PSU(2) invariant at a fifth root of unity for a 3-manifold whose first Z/5Z-Betti number is positive.

The linear skein theory for the Kauffman bracket was introduced by Lickorish [11,12]. It gives an elementary construction of quantum SU(2) invariant of 3-manifolds. In this paper we prove basic properties of the linear skein theory for quantum SU(3) invariant. By using them we give an elementary construction of quantum SU(3) invariant of 3-manifolds and prove topological invariance of the invariant along the construction.