Deposits of clastic carbonate-dominated (calciclastic) sedimentary slope systems in the rock record have been identified mostly as linearly-consistent carbonate apron deposits, even though most ancient clastic carbonate slope deposits fit the submarine fan systems better. Calciclastic submarine fans are consequently rarely described and are poorly understood. Subsequently, very little is known especially in mud-dominated calciclastic submarine fan systems. Presented in this study are a sedimentological core and petrographic characterisation of samples from eleven boreholes from the Lower Carboniferous of Bowland Basin (Northwest England) that reveals a >250 m thick calciturbidite complex deposited in a calciclastic submarine fan setting. Seven facies are recognised from core and thin section characterisation and are grouped into three carbonate turbidite sequences. They include: 1) Calciturbidites, comprising mostly of highto low-density, wavy-laminated bioclast-rich facies; 2) low-density densite mudstones which are characterised by planar laminated and unlaminated muddominated facies; and 3) Calcidebrites which are muddy or hyper-concentrated debrisflow deposits occurring as poorly-sorted, chaotic, mud-supported floatstones. These

Phd by thesis

* Introduction* Knots and links in $S^3$* Grid diagrams* Grid homology* The invariance of grid homology* The unknotting number and $\tau$* Basic properties of grid homology* The slice genus and $\tau$* The oriented skein exact sequence* Grid homologies of alternating knots* Grid homology for links* Invariants of Legendrian and transverse knots* The filtered grid complex* More on the filtered chain complex* Grid homology over the integers* The holomorphic theory* Open problems* Homological algebra* Basic theorems in knot theory* Bibliography* Index

Grid Homology for Knots and Links

We define invariants of null--homologous Legendrian and transverse knots in contact 3--manifolds. The invariants are determined by elements of the knot Floer homology of the underlying smooth knot. We compute these invariants, and show that they do not vanish for certain non--loose knots in overtwisted 3--spheres. Moreover, we apply the invariants to find transversely non--simple knot types in many overtwisted contact 3--manifolds.

Heegaard Floer invariants of Legendrian knots in contact three--manifolds

For a word w in an alphabet Γ, the alternation word digraph Alt(w), a certain directed acyclic graph associated with w, is presented as a means to analyze the free spectrum of the Perkins monoid $\mathbf{B_2^1}$
 . Let $(f_n^{\mathbf{B_2^1}})$
 denote the free spectrum of $\mathbf{B_2^1}$
 , let a
 n
 be the number of distinct alternation word digraphs on words whose alphabet is contained in {x
 1,..., x
 n
 }, and let p
 n
 denote the number of distinct labeled posets on {1,..., n}. The word problem for the Perkins semigroup $\mathbf{B_2^1}$
 is solved here in terms of alternation word digraphs: Roughly speaking, two words u and v are equivalent over $\mathbf{B_2^1}$
 if and only if certain alternation graphs associated with u and v are equal. This solution provides the main application, the bounds: $p_n \leq a_n \leq f_n^{\mathbf{B_2^1}} \leq 2^{n}a_{2n}^2$
 . A result of the second author in a companion paper states that $(\operatorname{log} \; a_n)\in O(n^3)$
 , from which it follows that $(\operatorname{log} f_n^{\mathbf{B_2^1}})\in O(n^3)$
 as well. Alternation word digraphs are of independent interest combinatorially. It is shown here that the computational complexity problem that has as instance {u,v} where u,v are words of finite length, and question “Is Alt(u) = Alt(v)?”, is co-NP-complete. Additionally, alternation word digraphs are acyclic, and certain of them are natural extensions of posets; each realizer of a finite poset determines an extension by an alternation word digraph.

Word Problem of the Perkins Semigroup via Directed Acyclic Graphs

Using the grid diagram formulation of knot Floer homology, Ozsvath, Szabo and Thurston defined an invariant of transverse knots in the tight contact 3‐sphere. Shortly afterwards, Lisca, Ozsvath, Stipsicz and Szabo defined an invariant of transverse knots in arbitrary contact 3‐manifolds using open book decompositions. It has been conjectured that these invariants agree where they are both defined. We prove this fact by defining yet another invariant of transverse knots, showing that this third invariant agrees with the two mentioned above. 57M27; 57R58

/pdf/on-the-equivalence-of-legendrian-and-transverse-invariants-303xdi72rj.pdf

On the equivalence of Legendrian and transverse invariants in knot Floer homology

In 1997, Chekanov gave the first example of a Legendrian nonsimple knot type: the $m(5_2)$ knot. Epstein, Fuchs, and Meyer extended his result by showing that there are at least $n$ different Legendrian representatives with maximal Thurston--Bennequin number of the twist knot $K_{-2n}$ with crossing number $2n+1$. In this paper we give a complete classification of Legendrian and transverse representatives of twist knots. In particular, we show that $K_{-2n}$ has exactly $\lceil\frac{n^2}2\rceil$ Legendrian representatives with maximal Thurston--Bennequin number, and $\lceil\frac{n}{2}\rceil$ transverse representatives with maximal self-linking number. Our techniques include convex surface theory, Legendrian ruling invariants, and Heegaard Floer homology.

Legendrian and transverse twist knots

https://www.ems-ph.org/journals/show_pdf.php?issn=1435-9855&vol=15&iss=3&rank=11

In this paper we extend the idea of bordered Floer homology to knots and links in $S^3$: Using a specific Heegaard diagram, we construct gluable combinatorial invariants of tangles in $S^3$, $D^3$ and $I\times S^2$. The special case of $S^3$ gives back a stabilized version of knot Floer homology.

/pdf/combinatorial-tangle-floer-homology-cpy0khpwdh.pdf

Combinatorial tangle Floer homology

The study of Legendrian and transverse knots is central in contact geometry. A Legendrian knot with a given knot type has two classical invariants: its Thurston–Bennequin number and its rotation number. The problem of classifying Legendrian knots up to Legendrian isotopy naturally leads to the question whether these invariants classify Legendrian knots. A knot type is called Legendrian simple if any two realizations of it with equal classical invariants are Legendrian isotopic. For transverse knots there is only one classical invariant, the self-linking number. Similarly, the knot types for which transverse realizations are classified by the self-linking number are called transversely simple. The unknot and then torus knots and the figure-eight knot were proved by Eliashberg and Fraser [4] and by Etnyre and Honda [7], respectively, to be both Legendrian and transversely simple. By constructing a new invariant for Legendrian knots, Chekanov [3] showed that not all knots are Legendrian simple; in particular he proved that the knot 52 is not Legendrian simple. Later many other Legendrian nonsimple knots were detected by Epstein, Fuchs and Meyer [5] and by Ng [13]. The case for transverse knots turned out to be harder. Birman and Menasco [1] and Etnyre and Honda [9] constructed families of transversely nonsimple knots using braid and convex surface theory. Recently Ng, Ozsvath and Thurston [14] gave such examples using the Legendrian invariant in knot Floer homology. Heegaard Floer homology b HF .Y /, HF .Y / defined by Ozsvath and Szabo [17] are invariants for three-manifolds. They extended the construction [16] to give the invariants 1 HFK.Y;K/, HFK .Y;K/ for null-homologous knots K Y via doubly pointed Heegaard diagrams. Using Heegaard diagrams with multiple basepoints they generalized the invariants for links [15]. Multiply pointed Heegaard diagrams turned out to be extremely useful in the case of knots as well, and led to the discovery of a

/pdf/transversely-nonsimple-knots-2c4muv6d2y.pdf

Transversely nonsimple knots

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Papers

Legendrian and transverse twist knots

On the equivalence of Legendrian and transverse invariants in knot Floer homology

Legendrian and transverse twist knots

Combinatorial tangle Floer homology

Transversely nonsimple knots