On 4-Move Equivalence Classes of Knots and Links of Two Components

Da̧bkowski, Mieczysław K.; Jablan, Slavik; Khan, Noureen; Sahi, R. K.

doi:10.1142/s0218216511008668

Cited by 8 publications

(2 citation statements)

References 6 publications

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“…alternating links with 12 crossings (cf. [1,16]). On the other hand, there is a growing belief that the problem is false.…”

Section: Figure 1 4-movementioning

confidence: 99%

The Dabkowski-Sahi invariant and $4$-moves for links

Miyazawa¹,

Wada²,

Yasuhara³

2020

Preprint

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Dabkowski and Sahi defined an invariant of a link in the 3-sphere, which is preserved under 4-moves. This invariant is a quotient of the fundamental group of the complement of the link. It is generally difficult to distinguish the Dabkowski-Sahi invariants of given links. In this paper, we give a necessary condition for the existence of an isomorphism between the Dabkowski-Sahi invariant of a link and that of the corresponding trivial link. Using this condition, we provide a practical obstruction to a link to be trivial up to 4-moves.Problem 1.1 (Dabkowski and Przytycki [3]). Is it true that two 2-component links with the same linking number are 4-equivalent?This problem has been shown to be true for several classes of 2-component links, e.g. 2-algebraic links, closed 3-braid links, links with 11 or less crossings, and

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“…alternating links with 12 crossings (cf. [1,16]). On the other hand, there is a growing belief that the problem is false.…”

Section: Figure 1 4-movementioning

confidence: 99%

The Dabkowski-Sahi invariant and $4$-moves for links

Miyazawa¹,

Wada²,

Yasuhara³

2020

Preprint

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“…The trefoil and the unknot are not pass move equivalents, therefore, the pass move is not an unknotting operation for knots. On the other hand, in [10] Dabkowski et al proved that all knots up to 12 crossings reduce to the trivial knot by 4-moves. They also showed that links of 2 components with at most 11 crossings reduce to either the trivial link or the Hopf link using a finite number of 4moves.…”

Section: Introductionmentioning

confidence: 99%

A new unknotting operation for classical and welded links

Ali¹,

Yang²,

Lei³

et al. 2022

Preprint

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An unknotting operation is a local change on knot diagrams. Any knot diagram can be transformed into a trivial unknot diagram by a series of unknotting operations plus some Reidemeister moves. Unknotting operations have significant importance in knot theory; they are used to study the complexity of knots and invariants of knots. Unknotting operations are also important in biology, chemistry, and physics to study the sophisticated entanglements of strings, organic compounds, and DNA. In this paper, we introduced a new unknotting operation called the diagonal move for classical and welded knots. We show that the crossing change, ∆-move, ♯-move, Γ-move, n-gon move, pass move, and 4-move can be realized by a sequence of diagonal moves. We also define the distance induced by the diagonal move and study its properties.

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