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

A spanning tree expansion of the jones polynomial

On knot Floer homology and lens space surgeries

Closed incompressible surfaces in alternating knot and link complements

This paper concentrates on the construction of invariants of knots, such as the Jones polynomials and the Vassiliev invariants, and the relationships of these invariants to other mathematics (such as Lie algebras).

An Introduction to Knot Theory

The classification of alternating links

A complete solution is given to the classification problem for oriented links which are closed three-braids. The Classification Theorem asserts that, up to a finite list of exceptional cases, links which can be represented by closed 3-braids are represented by a unique conjugacy class in the group of 3-braids. The exceptional cases are the expected ones (links of braid index 1 and 2) and an unexpected infinite family of invertible links, each member of which has two 3braid axes. The two axes correspond to diagrams which are related by \"braid-preserving flypes\". An algorithm is given which begins with an arbitrary closed 3-braid (or alternatively any link diagram with 3 Seifert circles), and converts it into a normal form which characterizes its oriented link type in oriented 3-space. One can decide from the normal form whether the link is prime or composite, split or irreducible, amphicheiral and or invertible. One can decide if the braid index is 3, 2 or 1. Using related results of P. J. Xu, one may determine the genus and construct a surface of maximum Euler characteristic with boundary the given link. It is proved that the stabilization index of a link which is represented by a closed 3-braid is < 1, i.e. any two 3-braid representatives of the same link type become conjugate after a single stabilization to B4.

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Studying links via closed braids. III. Classifying links which are closed 3-braids

We announce a proof of the Tait flyping conjecture; the confirmation of this conjecture renders almost trivial the problem of deciding whether two given alternating link diagrams represent equivalent links. The proof of the conjecture also shows that alternating links have no "hidden" symmetries. In the nineteenth century, the celebrated physicist and knot tabulator P. G. Tait proposed the following conjecture: given reduced, prime alternating diagrams D{, D2 of a knot (or link), it is possible to transform Dj to D2 by a sequence of flypes, where a flype is a transformation most easily described by the pictures of Figure 1 on p. 404. In performing a flype, the tangle represented by the shaded disc labelled SA is turned upside-down so that the crossing to its left is removed by untwisting, and a new crossing is created to its right; if the tangle diagram SA has no crossing, the flype leaves the link diagram unchanged up to isomorphism, whereas if the tangle diagram SB should have no crossing, the flype amounts merely to a rotation of the complete link diagram about an axis in the projection 2-sphere. During the last few years, some partial results have appeared; in particular it follows from the analysis of [B-S] on arborescent links that any two alternating diagrams of a link which are algebraic (i.e. which have Conway basic polyhedron 1*) must be related via a sequence of flypes. A slightly stronger version of this result is set forth in [T4], where the conclusion is obtained for a pair of alternating diagrams only one of which is given as algebraic. It follows from the results of [B-M] that the Tait conjecture holds for link diagrams which are closures of alternating 3-string braid diagrams. K. Murasugi and J. Przytycki [M-P] have proved a number of results on graph polynomials which have lent support to the conjecture. Very recently, A. Schrijver [S] has announced Received by the editors January 20, 1991. 1980 Mathematics Subject Classification (1985 Revision). Primary 57M25. The first author was partially supported by NSF Grant no. DMS 9002673. ©1991 American Mathematical Society 0273-0979/91 $1.00+ $.25 per page 403 404 W. W. MENASCO AND M. B. THISTLETHWAITE

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The Tait flyping conjecture

The main result of this paper is a negative answer to the question: are all transversal knot types transversally simple? An explicit infinite family of examples is given of closed 3-braids that define transversal knot types that are not transversally simple. The method of proof is topological and indirect.

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William W. Menasco

Papers

Closed incompressible surfaces in alternating knot and link complements

The classification of alternating links

Studying links via closed braids. III. Classifying links which are closed 3-braids

The Tait flyping conjecture

Stabilization in the braid groups II: Transversal simplicity of knots