On 4–fold covering moves

Apostolakis, Nikos

doi:10.2140/agt.2003.3.117

Cited by 8 publications

(15 citation statements)

References 11 publications

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“…arcs at a crossing the semiarcs We explain a bijection below (1). Denote by D o the unoriented link diagram D equipped with an orientation.…”

Section: Symmetric Quandles and Symmetric Coloringsmentioning

confidence: 99%

“…Conversely, given an X -coloring of D , if we restrict the X -coloring to the orientations of D o , then we have an X -coloring of D o as an inverse of the map (1). Hence the map (1) turns out to be bijective (see [18,Theorem 6.7] for detail).…”

Section: Symmetric Quandles and Symmetric Coloringsmentioning

confidence: 99%

“…Hence, we may consider any 3-manifold to be a labeled link. Further, it is known (see Apostolakis [1] and Bobtcheva and Piergallini [2]) that homeomorphism classes of 3-manifolds are in 1-1 correspondence with the set of labeled links modulo some "MI and MII moves" (see Figure 3). The key point is that, using these facts, she presented the Dijkgraaf-Witten invariants of 3-manifolds as some invariants of labeled links.…”

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4–fold symmetric quandle invariants of 3–manifolds

Nosaka

2011

Algebr. Geom. Topol.

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The paper introduces 4-fold symmetric quandles and 4-fold symmetric quandle homotopy invariants of 3-manifolds. We classify 4-fold symmetric quandles and investigate their properties. When the quandle is finite, we explicitly determine a presentation of its inner automorphism group. We calculate the container of the 4-fold symmetric quandle homotopy invariant. We also discuss symmetric quandle cocycle invariants and coloring polynomials of 4-fold symmetric quandles. 57M12, 57M25, 57M27, 57N70, 58K65; 55Q52, 22A30, 11E57, 55R40, 05E15 IntroductionA quandle is a set with a certain binary operation satisfying a self-distribute law. Quandles are adapted to the oriented link theory. For an oriented link L S 3 , Joyce [16] defined the link quandle Q L as an analog of the fundamental group 1 .S 3 n L/. For a quandle X , a quandle homomorphism Q L ! X is called an X -coloring of L. From algebraic topology, given a quandle X , Fenn, Rourke and Sanderson [9] defined the rack space analogous to the classifying space of groups. They [10; 11] show that the second homotopy group is isomorphic to a bordism group consisting of all "framed X -colorings". Then, a quandle homotopy invariant of oriented links can be defined by an invariant valued in the group ring ZOE 2 .BX /, where the space BX is a certain modification of the rack space. On the other hand, quandle cocycle invariants of oriented links introduced by Carter et al [3] using 2-cocycles of H 2 .BX I A/ are computable and practical; they can, however, be derived from the quandle homotopy invariant (see, eg, Carter, Kamada and Saito [4] and Fenn and Rourke [8]).In another direction, Hatakenaka [13] reformulated certain Dijkgraaf-Witten invariants of 3-manifolds [5] as quandle cocycle invariants. To see this, she made use of the fact that any 3-manifold is a 4-fold simple branched covering of the 3-sphere branched along a link L. Then the associated simple monodromy representation onto S 4 can be regarded as an S -coloring of L, which we call a labeled link. Here S WD f.ij / 2 S 4 g is a quandle with the conjugate operation. Hence, we may consider any 3-manifold to be a labeled link. Further, it is known (see Apostolakis [1] and Bobtcheva and Piergallini [2]) that homeomorphism classes of 3-manifolds are in 1-1 correspondence with the set of labeled links modulo some "MI and MII moves" (see Figure 3). The key point is that, using these facts, she presented the Dijkgraaf-Witten invariants of 3-manifolds as some invariants of labeled links.In this paper, our purpose is to construct and study an invariant of 3-manifolds obtained from the quandle homotopy invariant of labeled links. The idea behind the construction is simple: since any 3-manifold can be regarded as a labeled link of L, we define a quandle X over S , and consider X -colorings obtained by lifting the S -coloring to be an invariant of the 3-manifold. For this, noting that the monodromies are unrelated to orientations of links, we focus on symmetric quandles introduced by Kamada [17] which are suitable for un...

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“…arcs at a crossing the semiarcs We explain a bijection below (1). Denote by D o the unoriented link diagram D equipped with an orientation.…”

Section: Symmetric Quandles and Symmetric Coloringsmentioning

confidence: 99%

Section: Symmetric Quandles and Symmetric Coloringsmentioning

confidence: 99%

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confidence: 99%

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4–fold symmetric quandle invariants of 3–manifolds

Nosaka

2011

Algebr. Geom. Topol.

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“…For example, this is the case of one of the wellknown Montesinos moves (cf. [18], [20], [1] or [3]) for simple coverings of S 3 branched over a link. Such crossing change has already been used in the construction of universal links (cf.…”

Section: Some Covering Movesmentioning

confidence: 99%

A universal ribbon surface in $B^{4}$

Piergallini

Zuddas

2005

Proc. London Math. Soc.

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“…This was proved to be true by the second author in [62], where the question was also posed, whether these local moves together with stabilization also suffice for simple coverings of arbitrary degree. In [4] Apostolakis answered this question in the positive for 4-fold coverings, up to 5-fold stabilization.…”

Section: Introductionmentioning

confidence: 99%

On 4-Dimensional 2-Handlebodies and 3-Manifolds

Bobtcheva

Piergallini

2012

J. Knot Theory Ramifications

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We show that for any n ≥ 4 there exists an equivalence functor [Formula: see text] from the category [Formula: see text] 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 [Formula: see text] 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 [Formula: see text], where [Formula: see text] is an equivalence functor to a universal braided category [Formula: see text] freely generated by a Hopf algebra object H. In this way, we get a complete algebraic description of the category [Formula: see text]. From this we derive an analogous description of the category [Formula: see text] of 2-framed relative 3-dimensional cobordisms, which resolves a problem posed by Kerler.

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On 4–fold covering moves

Cited by 8 publications

References 11 publications

4–fold symmetric quandle invariants of 3–manifolds

4–fold symmetric quandle invariants of 3–manifolds

A universal ribbon surface in $B^{4}$

On 4-Dimensional 2-Handlebodies and 3-Manifolds

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