The purpose of this note is to announce a new isotopy invariant of oriented links of tamely embedded circles in 3-space.We represent links by plane projections, using the customary conventions that the image of the link is a union of transversely intersecting immersed curves, each provided with an orientation, and undercrossings are indicated by broken lines. Following Conway [6], we use the symbols L+, Lo, L_ to denote links having plane projections which agree except in a small disk, and inside that disk are represented by the pictures of Figure 1.Conway showed that the one-variable Alexander polynomials of L+, Lo, L_ (when suitably normalized) satisfy the relation It was evident from the circumstances that the four groups arrived at their results completely independently of each other, although all were inspired by the work of Jones (cf. [10], and also [8, 9]). The degree of simultaneity was such that, by common consent, it was unproductive to try to assess priority. Indeed it would seem that there is enough credit for all to share in.Each of these papers was refereed, and we would have happily published any one of them, had it been the only one under consideration. Because the alternatives of publication of all four or of none were both unsatisfying, all have agreed to the compromise embodied here of a paper carrying all six names as coauthors, consisting of an introductory section describing the basics written by a disinterested party, and followed by four sections, one written by each of the four groups, briefly describing the highlights of their own approach and elaboration.

The protein topology database KnotProt, http://knotprot.cent.uw.edu.pl/, collects information about protein structures with open polypeptide chains forming knots or slipknots. The knotting complexity of the cataloged proteins is presented in the form of a matrix diagram that shows users the knot type of the entire polypeptide chain and of each of its subchains. The pattern visible in the matrix gives the knotting fingerprint of a given protein and permits users to determine, for example, the minimal length of the knotted regions (knot's core size) or the depth of a knot, i.e. how many amino acids can be removed from either end of the cataloged protein structure before converting it from a knot to a different type of knot. In addition, the database presents extensive information about the biological functions, families and fold types of proteins with non-trivial knotting. As an additional feature, the KnotProt database enables users to submit protein or polymer chains and generate their knotting fingerprints.

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While analyzing all available protein structures for the presence of knots and slipknots, we detected a strict conservation of complex knotting patterns within and between several protein families despite their large sequence divergence. Because protein folding pathways leading to knotted native protein structures are slower and less efficient than those leading to unknotted proteins with similar size and sequence, the strict conservation of the knotting patterns indicates an important physiological role of knots and slipknots in these proteins. Although little is known about the functional role of knots, recent studies have demonstrated a proteinstabilizing ability of knots and slipknots. Some of the conserved knotting patterns occur in proteins forming transmembrane channels where the slipknot loop seems to strap together the transmembrane helices forming the channel.protein knots | knot theory | topology T here are increasing numbers of known proteins that form linear open knots in their native folded structure (1, 2). In general, knots in proteins are orders of magnitude less frequent than would be expected for random polymers with similar length, compactness, and flexibility (3). In principle, the polypeptide chains folding into knotted native protein structures encounter more kinetic difficulties than unknotted proteins (4-12). Therefore, it is believed that knotted protein structures were, in part, eliminated during evolution because proteins that fold slowly and/or nonreproducibly should be evolutionarily disadvantageous for the hosting organisms. Nevertheless, there are several families of proteins that reproducibly form simple knots, complex knots, and slipknots (1, 2). In these proteins, the disadvantage of less efficient folding may be balanced by a functional advantage connected with the presence of these knots. Numerous experimental (13-16) and theoretical (17-27) studies have been devoted to understanding the precise nature of the structural and functional advantages created by the presence of these knots in protein backbones. It has been proposed that in some cases the protein knots and slipknots provide a stabilizing function that can act by holding together certain protein domains (4). In the majority of cases, however, one is unable to determine the precise structural and functional advantages provided by the presence of knots.To shed light on the function and formation of protein knots, we performed a thorough characterization of knotting within protein structures deposited in the Protein Data Bank (PDB) (28) by creating a precise mapping of the position and size of the knotted and slipknotted domains and knot tails (1, 2). We identified the types of knots that are formed by the backbone of the entire polypeptide chain and also by all continuous backbone portions of a given protein (1, 2). To characterize the knotting of proteins with linear backbones, one needs to set aside the orthodox rule of knot theory that states that all linear chains are unknotted because, by a continuous deformation,...