Width is not additive

Blair, Ryan; Tomova, Maggy

doi:10.2140/gt.2013.17.93

Cited by 18 publications

(37 citation statements)

References 25 publications

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“…This result is generalized in [5], where it was shown that any other bridge surface of such a knot must also have high genus or a high number of marked points. Several other recent constructions of interesting examples rely on such knots -in [2] Blair and Tomova construct knots for which width is not additive, and in [4] Johnson and Tomova construct examples of knots with two different bridge surfaces that require a high number of stabilizations and perturbations to become isotopic.…”

Section: Introductionmentioning

confidence: 99%

High distance bridge surfaces

Blair

Tomova

Yoshizawa

2013

Algebr. Geom. Topol.

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Abstract. Given integers b, c, g, and n, we construct a manifold M containing a c-component link L so that there is a bridge surface Σ for (M, L) of genus g that intersects L in 2b points and has distance at least n. More generally, given two possibly disconnected surfaces S and S ′ , each with some even number (possibly zero) of marked points, and integers b, c, g, and n, we construct a compact, orientable manifold M with boundary S ∪ S ′ such that M contains a c-component tangle T with a bridge surface Σ of genus g that separates ∂M into S and S ′ , |T ∩ Σ| = 2b and T intersects S and S ′ exactly in their marked points, and Σ has distance at least n.

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Section: Introductionmentioning

confidence: 99%

High distance bridge surfaces

Blair

Tomova

Yoshizawa

2013

Algebr. Geom. Topol.

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“…(2) There exist two knots K and K ′ in Figure 4 Note that a knot K in Figure 4 is not in a thin position, thus we have γ ∈ M CP (K)\T P (K). Moreover, we remark that there exists a knot K = K α in [8] such that T P (K) ∩ M CP (K) = ∅.…”

Section: Several Versions Of Thin Positionmentioning

confidence: 99%

“…Example 3.2. For a knot K ∈ K given in Theorem 1.3, by [8], there exists γ ∈ K in Figure 3 such that γ ∈ T P (K)\M CP (K) and γ is not in OT P (K) since there exists an embedding of K with two thick levels of width 10 and two thick levels of width 8. Since it is shown in [8] that every γ ∈ T P (K) has exactly three thick levels, each of width 10, we have pro(K) = 10 3 1 2 × 11 = 0.1515 .…”

Section: Several Versions Of Thin Positionmentioning

confidence: 99%

Height, trunk and representativity of knots

Blair¹,

Ozawa²

2019

J. Math. Soc. Japan

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In this paper, we investigate three geometrical invariants of knots, the height, the trunk and the representativity.First, we give a conterexample for the conjecture which states that the height is additive under connected sum of knots. We also define the minimal height of a knot and give a potential example which has a gap between the height and the minimal height.Next, we show that the representativity is bounded above by a half of the trunk. We also define the trunk of a tangle and show that if a knot has an essential tangle decomposition, then the representativity is bounded above by half of the trunk of either of the two tangles.Finally, we remark on the difference among Gabai's thin position, ordered thin position and minimal critical position. We also give an example of a knot which bounds an essential non-orientable spanning surface, but has arbitrarily large representativity.

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“…In this study I focused on constructing and making practical use of an axiomatic basis for MHA from biological considerations alone. Nonetheless, it will be interesting to study whether there are applicable theoretical treatments of low-dimensional topological objects, such as from mathematical knot theory [e.g., Blair and Tomova (2013)] or from models of quantum gravity in physics [e.g., Carlip (2017)]. If so there may exist additional mathematical tools that can be adapted for the alignment of two dimensional biological sequence homology.…”

Section: Mha Widths Are Not Additivementioning

confidence: 99%

Maximal homology alignment: A new method based on two-dimensional homology

Erives

2019

Preprint

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Maximal homology alignment is a new biologically-relevant approach to DNA sequence alignment that embraces microparalogy. It departs from the current method of gapped alignment, which uses a simple, binary state model of nucleotide position. In gapped alignment nucleotide positions have either no relationship (1-to-None) or else orthological relationship (1-to-1) with nucleotides in other sequences. Maximal homology alignment, however, allows additional states such as 1-to-Many and Many-to-Many, thus modeling both orthological and paralogical relationships, which together comprise the main homology types. Maximal homology alignment collects dispersed microparalogy into the same alignment columns on multiple rows, and thereby generates a two-dimensional representation of a single sequence. Sequence alignment then proceeds as the alignment of two-dimensional topological objects. The operations of producing and aligning two dimensional auto-alignments motivate a need for tests of two dimensional homological integrity that are both necessary and sufficient. Here, I work out and implement basic principles for computationally handling the two dimensions of positional homology, which are inherent to biological sequences due to replication slippage and related errors. I then show that maximal homology alignment is more suited and more informative than gapped alignment for modeling the evolution of genetic sequences, which harbor large numbers of small insertions and deletions originating as local microparalogy. These results show that both conserved and non-conserved genomic sequences are imbued with a signature of replication slippage that is absent from their random permutations. KEYWORDS dimensionality; DNA microfoam; Drosophila; enhancer DNAs; gapped alignment (GA); maximal homology alignment (MHA); positional homology; replication slippage; tandem repeats S equences of covalently-linked nucleotides and their evolutionary relationships are the basis of gene homology. The smallest possible scale to ascribe such homology is that of individual nucleotide positions ("site positional homology"). Of course this is possible only in the context of homological inference based on multiple nucleotide positions in a sequence.By studying the function and evolution of regulatory DNA sequences (transcriptional enhancers in Ciona and Drosophila), which are not constrained by rigid, protein-coding, triplet reading frames (Erives et al. Brittain et al. 2014;Stroebele and Erives 2016), I conclude that site positional homology is incorrectly handled by gapped alignment (GA). GA was originally adopted by necessity in order to compare protein sequences of divergent lengths (Braunitzer's gappy comparisons of α-and β-hemoglobin

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Width is not additive

Cited by 18 publications

References 25 publications

High distance bridge surfaces

High distance bridge surfaces

Height, trunk and representativity of knots

Maximal homology alignment: A new method based on two-dimensional homology

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