On the Classification of Rational 3-Tangles

Cabrera-Ibarra, Hugo

doi:10.1142/s021821650300286x

Cited by 19 publications

(20 citation statements)

References 8 publications

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“…Let [d 1 , d 2 ] be the segment between d 1 and d 2 in Γ • * . Now, we choose a homeomorphism 6 . This implies that δ and Γ • * have a bigon in Σ 0,6 .…”

Section: Now Consider Orientation Preserving Homeomorphismsmentioning

confidence: 99%

An algorithm to classify rational 3-tangles

Kwon

2015

J. Knot Theory Ramifications

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A 3-tangle T is the disjoint union of 3 properly embedded arcs in the unit 3-ball; it is called rational if there is a homeomorphism of pairs from (B 3 , T ) to (D 2 × I, {x 1 , x 2 , x 3 } × I). Two rational 3-tangles T and T ′ are isotopic if there is an orientationpreserving self-homeomorphism h : (B 3 , T ) → (B 3 , T ′ ) that is the identity map on the boundary. In this paper, we give an algorithm to check whether or not two rational 3tangles are isotopic by using a modified version of Dehn's method for classifying simple closed curves on surfaces.

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“…Let [d 1 , d 2 ] be the segment between d 1 and d 2 in Γ • * . Now, we choose a homeomorphism 6 . This implies that δ and Γ • * have a bigon in Σ 0,6 .…”

Section: Now Consider Orientation Preserving Homeomorphismsmentioning

confidence: 99%

An algorithm to classify rational 3-tangles

Kwon

2015

J. Knot Theory Ramifications

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“…To prove this, we repeatedly use the three axioms to remove the crossings below the dotted line of the diagram (a) and the diagram (b) respectively to have This completes the proof. H. Cabrera-Ibarra [3] defined the bracket polynomial of the rational 3-tangle diagram T of…”

Section: A New Invariant Of Rational 2-tanglesmentioning

confidence: 99%

“…H. Cabrera-Ibarra [3] found a pair of invariants which is defined for all rational 3-tangles. Each invariant is a 3 × 3 matrix with complex number entries.…”

Section: Introductionmentioning

confidence: 99%

Uniqueness of reduced alternating rational 3-tangle diagrams

Kwon

2015

J. Knot Theory Ramifications

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Tangles were introduced by J. Conway. In 1970, he proved that every rational 2-tangle defines a rational number and two rational 2-tangles are isotopic if and only if they have the same rational number. So, from Conway's result we have a perfect classification for rational 2-tangles. However, there is no similar theorem to classify rational 3-tangles.In this paper, we introduce an invariant of rational n-tangles which is obtained from the Kauffman bracket. It forms a vector with Laurent polynomial entries. We prove that the invariant classifies the rational 2-tangles and the reduced alternating rational 3-tangles. We conjecture that it classifies the rational 3-tangles as well.

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“…Equations (2)- (4) correspond to three product equations modeling the three in cis deletion experiments, while Equations (5)- (7) correspond to the three product equations modeling the three in cis inversion experiments. Equation (8) corresponds to the unknotted substrate equation for the two in trans experiments, while Equations (9) and (10) correspond to the product equations modeling in trans deletion and in trans inversion, respectively. In addition to modeling experimental results in [38], these equations also model results in [39; 59; 60].…”

Section: Normal Formmentioning

confidence: 99%

Tangle analysis of difference topology experiments: Applications to a Mu protein-DNA complex

Darcy

Luecke

Vázquez

2009

Algebr. Geom. Topol.

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We develop topological methods for analyzing difference topology experiments involving 3-string tangles. Difference topology is a novel technique used to unveil the structure of stable protein-DNA complexes. We analyze such experiments for the Mu protein-DNA complex. We characterize the solutions to the corresponding tangle equations by certain knotted graphs. By investigating planarity conditions on these graphs we show that there is a unique biologically relevant solution. That is, we show there is a unique rational tangle solution, which is also the unique solution with small crossing number. 57M25, 92C40 IntroductionBacteriophage Mu is a virus that infects bacteria. Mu transposase is involved in transposing the Mu genome within the DNA of the bacterial host. In [38], Pathania et al determined the shape of DNA bound within the Mu transposase protein complex using an experimental technique called difference topology; see Harshey and Jayaram [26;30], Kilbride et al [31], Grainge et al [24], Pathania et al [38;39] and Yin et al [59;60]. Their conclusion was based on the assumption that the DNA is in a branched supercoiled form as described near the end of Section 1 (see Figures 9 and 25). We show that this restrictive assumption is not needed, and in doing so, conclude that the only biologically reasonable solution for the shape of DNA bound by Mu transposase is the one found in [38] (shown in Figure 1). We will call this 3-string tangle the PJH solution. The 3-dimensional ball represents the protein complex, and the arcs represent the bound DNA. The Mu-DNA complex modeled by this tangle is called the Mu transpososome.The topological structure of the Mu transpososome will be a key ingredient in determining its more refined molecular structure and in understanding the basic mechanism crystallographic data is combined with the known Tn3 resolvase topological mechanisms of binding and strand-exchange to piece together a more detailed geometrical picture of two resolvase-DNA complexes (Sin and ı ).In Section 1 we provide some biological background and describe eight difference topology experiments from [38]. In Section 2, we translate the biological problem of determining the shape of DNA bound by Mu into a mathematical model. The mathematical model consists of a system of ten 3-string tangle equations ( Figure 11). Using 2-string tangle analysis, we simplify this to a system of four tangle equations ( Figure 24). In Section 3 we characterize solutions to these tangle equations in terms of knotted graphs. This allows us to exhibit infinitely many different 3-string tangle solutions. The existence of solutions different from the PJH solution raises the possibility of alternate acceptable models. In Sections 3-5, we show that all solutions to the mathematical problem other than the PJH solution are too complex to be biologically reasonable, where the complexity is measured either by the rationality or by the minimal crossing number of the 3-string tangle solution.In Section 3, we show that the only rational solution i...

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On the Classification of Rational 3-Tangles

Cited by 19 publications

References 8 publications

An algorithm to classify rational 3-tangles

An algorithm to classify rational 3-tangles

Uniqueness of reduced alternating rational 3-tangle diagrams

Tangle analysis of difference topology experiments: Applications to a Mu protein-DNA complex

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