On a certain move generating link-homology

Murakami, Hitoshi; Nakanishi, Yasutaka

doi:10.1007/bf01443506

Cited by 183 publications

(171 citation statements)

References 9 publications

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“…There are many prototypes for this in the recent literature. One example is the result of Murakami-Nakanishi [17], which is closely related to results of Matveev [15], which characterizes -equivalence classes of links in terms of their linking matrices. Another is the result of Habiro [13] classifying knots, all of whose finite-type invariants up to a certain degree are equal, via surgery along tree claspers.…”

Section: T S T T T Smentioning

confidence: 95%

Surgery presentations of coloured knots and of their covering links

Kricker

Moskovich

2009

Algebr. Geom. Topol.

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We consider knots equipped with a representation of their knot groups onto a dihedral group D 2n (where n is odd). To each such knot there corresponds a closed 3-manifold, the (irregular) dihedral branched covering space, with the branching set over the knot forming a link in it. We report a variety of results relating to the problem of passing from the initial data of a D 2n -coloured knot to a surgery presentation of the corresponding branched covering space and covering link. In particular, we describe effective algorithms for constructing such presentations. A by-product of these investigations is a proof of the conjecture that two D 2n -coloured knots are related by a sequence of surgeries along˙1-framed unknots in the kernel of the representation if and only if they have the same coloured untying invariant (a Z n -valued algebraic invariant of D 2n -coloured knots). 57M12; 57M25

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Section: T S T T T Smentioning

confidence: 95%

Surgery presentations of coloured knots and of their covering links

Kricker

Moskovich

2009

Algebr. Geom. Topol.

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“…The self C 2 -equivalence coincides with the self -equivalence, which is an equivalence relation generated by self -moves. A -move is a local move as illustrated in Figure 1; see Murakami and Nakanishi [15]. The -move is called a self -move if all strands in Figure 1 belong to the same component of a (string) link; see Shibuya [19].…”

Section: Introductionmentioning

confidence: 99%

Classification of string links up to self delta-moves and concordance

Yasuhara

2009

Algebr. Geom. Topol.

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“…In other words, one can undo L by a sequence of double crossing changes shown below in Figure 3, where the loop g g p + g Figure 3: A double crossing change is nullhomotopic. These double crossing changes can be achieved by surgery on Y-graphs whose leaves are nullhomotopic, see [11,12] and also Lemma 4.7 below. So far, each of the Y-graphs have two leaves that bound a disk that intersects L at most once and a nullhomotopic leaf.…”

Section: Lemma 45 [3 Lemma 53] Let T Be a Trivial N-componentmentioning

confidence: 99%

Homology surgery and invariants of 3–manifolds

Garoufalidis

Levine

2001

Geom. Topol.

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We introduce a homology surgery problem in dimension 3 which has the property that the vanishing of its algebraic obstruction leads to a canonical class of π -algebraically-split links in 3-manifolds with fundamental group π . Using this class of links, we define a theory of finite type invariants of 3-manifolds in such a way that invariants of degree 0 are precisely those of conventional algebraic topology and surgery theory. When finite type invariants are reformulated in terms of clovers, we deduce upper bounds for the number of invariants in terms of π -decorated trivalent graphs. We also consider an associated notion of surgery equivalence of π -algebraically split links and prove a classification theorem using a generalization of Milnor'sμ-invariants to this class of links.

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On a certain move generating link-homology

Cited by 183 publications

References 9 publications

Surgery presentations of coloured knots and of their covering links

Surgery presentations of coloured knots and of their covering links

Classification of string links up to self delta-moves and concordance

Homology surgery and invariants of 3–manifolds

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