generalizing the hough transform to detect arbitrary shapes abstract the hough transform is a method for detecting curves by exploiting the duality between points on a curve and parameters of that curve the initial work showed how to detect both analytic curves 1 2 and non analytic curves 3 but these methods were restricted to binary edge images this work was generalized to the detection of some analytic curves in grey level images specifically lines 4 circles 5 and parabolas 6 the line detection case is the best known of these and has been ingeniously exploited in several applications 7 8 9 we show how the boundaries of an arbitrary non analytic shape can be used to construct a mapping between image space and hough transform space such a mapping can be exploited to detect instances of that particular shape in an image furthermore variations in the shape such as rotations scale changes or figure ground reversals correspond to straightforward transformations of this mapping however the most remarkable property is that such mappings can be composed to build mappings for complex shapes from the mappings of simpler component shapes this makes the generalized hough transform a kind of universal transform which can be used to find arbitrarily complex shapes

invariant character recognition with zernike and orthogonal fourier mellin moments abstract in this paper we consider the use of orthogonal moments for invariant classification of alphanumeric characters of different size in addition to the zernike and pseudo zernike moments zms and pzms which have been previously proposed for invariant character recognition a new method of combining orthogonal fourier mellin moments ofmms with centroid bounding circle scaling is introduced which is shown to be useful in characterizing images with large variability through extensive experimentation using zms and ofmms as features different scaling methodologies and classifiers it is shown that ofmms give the best overall performance in terms of both image reconstruction and classification accuracy

on the quantum filtration of the khovanov rozansky cohomology abstract we prove the quantum filtration on the khovanov rozansky link cohomology h p with a general degree n 1 monic potential polynomial p x is invariant under reidemeister moves and construct a spectral sequence converging to h p that is invariant under reidemeister moves whose e 1 term is isomorphic to the khovanov rozansky sl n cohomology h n then we define a generalization of the rasmussen invariant and study some of its properties we also discuss relations between upper bounds of the self linking number of transversal links in standard contact s 3

template matching using the parametric template vector with translation rotation and scale invariance a new method for the template matching invariant to image translation rotation and scaling is proposed in the first step of our approach the ring projection transform rpt process is used to convert the 2d template in a circular region into a 1d gray level signal as a function of radius the advantages of the rpt process are that it owns the characteristic of rotation invariance and reduces the computational complexity of normalized correlation nc then the template matching is performed by constructing a parametric template vector ptv of the 1d gray level signal with differently scaled templates of the object the merits of our approach are that it not only obtains rotation invariance but also scale invariance additionally our approach is conceptually simple easy to implement and only one training image is needed for the training phase experimental results show that the computational time of the proposed approach is faster and the performance is better than the parametric template pt method and the affine moment invariants amis method in the image rotation or scaling moreover our approach not only enjoys high accurate rate under the changes of translation rotation and scale but also estimates the scaling value of the target object in the input scene experiments with gaussian noise demonstrate that the proposed algorithm is robust to detect the target object with the changes of translation orientation and scale this indicates that our approach is suitable for on line template matching with scene translation rotation and scaling

image analysis by bessel fourier moments in this paper we proposed a new set of moments based on the bessel function of the first kind named bessel fourier moments bfms which are more suitable than orthogonal fourier mellin and zernike moments for image analysis and rotation invariant pattern recognition compared with orthogonal fourier mellin and zernike polynomials of the same degree the new orthogonal radial polynomials have more zeros and these zeros are more evenly distributed the bessel fourier moments can be thought of as generalized orthogonalized complex moments theoretical and experimental results show that the bessel fourier moments perform better than the orthogonal fourier mellin and zernike moments ofmms and zms in terms of image reconstruction capability and invariant recognition accuracy in noise free noisy and smooth distortion conditions

an invariant of regular isotopy this paper studies a two variable laurent polynomial invariant of regular isotopy for classical unoriented knots and links this invariant is denoted lk for a link k and it satisfies the axioms 1 regularly isotopic links receive the same polynomial 2 lo0 1 3 l al l a l 4 l small diagrams indicate otherwise identical parts of larger diagrams regular isotopy is the equivalence relation generated by the reidemeister moves of type ii and type iii invariants of ambient isotopy are obtained from l by writhe normalization

a novel two stage template matching method for rotation and illumination invariance abstract template matching is the process of determining the presence and the location of a reference image or an object inside a scene image under analysis by a spatial cross correlation process conventional cross correlation type algorithms are computationally expensive furthermore when the object in the image is rotated the conventional algorithms cannot be used for practical purposes in this paper an algorithm for a rotation invariant template matching method based on the combination of the projection method and zernike moments is proposed the algorithm consists of two stages in the first stage the matching candidates are selected using a computationally low cost feature frequency domain calculation was adopted to reduce the computational cost for this stage in the second stage rotation invariant template matching is performed only on the matching candidates using zernike moments

orthogonal moments based on exponent functions exponent fourier moments abstract in this paper we propose a new set of orthogonal moments based on exponent functions named exponent fourier moments efms which are suitable for image analysis and rotation invariant pattern recognition compared with zernike polynomials of the same degree the new radial functions have more zeros and these zeros are evenly distributed this property make efms have strong ability in describing image unlike zernike moments the kernel of computation of efms is extremely simple theoretical and experimental results show that exponent fourier moments perform very well in terms of image reconstruction capability and invariant recognition accuracy in noise free noisy and smooth distortion conditions the exponent fourier moments can be thought of as generalized orthogonal complex moments

splitting words and lorenz braids abstract a canonical way to split lorenz periodic orbits is presented this allows one to recognize the braids on the lorenz template which can occur in non chaotic semi flows on this surface we first present a combinatorial version of our main result and then merely reinterpret it in topological terms

sofic systems and encoding data techniques of symbolic dynamics are applied to prove the existence of codes suitable for certain input restricted channels this generalizes the earlier work of adler coppersmith and hassner on the same problem

on the unknotting number of minimal diagrams answering negatively a question of bleiler we give examples of knots where the difference between minimal and maximal unknotting number of minimal crossing number diagrams grows beyond any extent

scale invariant template matching using histogram of dominant gradients abstract this paper presents a histogram based template matching method that copes with the large scale difference between target and template images most of the previous template matching methods are sensitive to the scale difference between target and template images because the features extracted from the images are changed according to the scale of the images to overcome this limitation we introduce the concept of dominant gradients and describe an image as the feature that is tolerant to scale changes to this end we first extract the dominant gradients of a template image and represent the template image as the grids of histograms of the dominant gradients then the arbitrary regions of a target image with various locations and scales are matched with the template image via histogram matching experimental results show that the proposed method is more robust to scale difference than previous template matching techniques

braids and the jones polynomial an important new invariant of knots and links is the jones polynomia and the subsequent generalized jones polynomial or two variable polynomial we prove inequalities relating the number of strands and the crossing number of a braid with the exponents of the variables in the generalized jones polynomial which is associated to the link formed from the braid by connecting the bottom ends to the top ends we also relate an exponent in the polynomial to the number of components of this link in j vaughan jones introduced a new polynomial invariant of oriented knots and links in 3 space subsequently a number of researchers freyd and yetter hoste lickorish and millett and ocneanu see fyhlmo independently realized that thlis could be generalized to produce an invariant which is a laurent polynomial of two variables and which specializes to give both the invariant of j and the classical alexander polynomial see r more precisely there is a function p from isotopy classes of oriented links to z x x 1 y y l if l is an oriented link we will write p l j x y by abuse of notation we will also write p b for b an oriented braid meaning of course the invariant associated to the oriented link determined by closing up b in the usual way several equivalent ways of codifying this invariant are described in fyhlmo we have chosen the approach of lickorish and millett hence our p k j x y is precisely their pk with x and y substituted for i and m respectively see fyhlmo and l m there appears as yet to be no consensus on a name for this two variable invariant in order to avoid awkward circumlocutions and being unable to amalgamate the names of the six or eight individuals who actually discovered it we have chosen to refer to it as the generalized jones polynomial or jones polynomial for short when there is no likelihood of confusion with the original jones polynomial this polynomial which we will denote j x y is characterized by figure 1 the interpretation of this is as follows given a regular projection of a link k with a crossing pictured as below one can form two new links k and ko which are identical to k except the one crossing is changed as shown when this is done the jones polynomials of the three links are related as in the formula it is a remarkable fact that this together with the normalization that p unknot 1 un quely determines a laurent polynomial j x y and that this polynomial is a link invariant i e is independent of the projection this result can be found in l m this formula in fact gives a good method for recursively computing j x y one received by the editors january 13 1986 1980 mathematics subject classification 1985 revision primary 57m25 research supported in part by nsf grant mcs 83

on the generating sequences of regular languages on k symbols the main result is a characterization of the generating sequences of the length of words in a regular language on k symbols we say that a sequence s of integers is regular if there is a finite graph g with two vertices i t such that sn is the number of paths of length n from i to t in g thus the generating sequence of a regular language is regular we prove that a sequence s is the generating sequence of a regular language on k symbols if and only if both sequences s sn n 0 and t kn sn n 0 are regular

circuit partitions and the homfly polynomial of closed braids we present an expansion of the homfly polynomial p d z a of a braid diagram d in terms of its circuit partitions another aspect of this result is an expression of p d z a as the trace of a matrix associated to d in a simple way we show how certain degree properties of the homfly polynomial can be derived easily from this model in particular we obtain that if d is a positive braid diagram on n strings with w crossings the maximum degree of p d z a in the variable a equals n 1 w resume nous presentons une expansion pour le polyn6me homfly p d z a d un diagramme de tresse d en termes de ses partitions en circuits un autre aspect de ce resultat consiste en une expression de p d z a comme trace d une matrice associee de facon simple a d nous montrons comment certaines proprietes de degre du polyn6me homfly derivent simplement de ce modele en particulier nous obtenons que pour un diagramme de tresse positif d a n brins et uw croisements le degre maximum de p d z a en la variable a est egal a n 1 w

recognitive aspects of moment invariants moment invariants are evaluated as a feature space for pattern recognition in terms of discrimination power and noise tolerance the notion of complex moments is introduced as a simple and straightforward way to derive moment invariants through this relation properties of complex moments are used to characterize moment invariants aspects of information loss suppression and redundancy encountered in moment invariants are investigated and significant results are derived the behavior of moment invariants in the presence of additive noise is also described

pattern recognition with moment invariants a comparative study and new results abstract moment invariants have been proposed as pattern sensitive features in classification and recognition applications in this paper the authors present a comprehensive study of the effectiveness of different moment invariants in pattern recognition applications by considering two sets of data handwritten numerals and aircrafts the authors also present a detailed study of zernike and pseudo zernike moment invariants including a new procedure for deriving the moment invariants in addition the authors introduce a new normalization scheme that reduces the large dynamic range of these invariants as well as implicit redundancies in these invariants based on a comprehensive study with both handwritten numerals and aircraft data the authors show that the new method of deriving zernike moment invariants along with the new normalization scheme yield the best overall performance even when the data are degraded by additive noise

rotation invariant pattern matching with color ring projection in this paper we propose a rotation invariant pattern matching scheme for detecting objects in complex color images the complexity and computational load for matching colored objects in arbitrary orientations are reduced significantly by the 1 d color ring projection representation it can rapidly select the possible locations of a reference template in the input scene by computing the normalized correlation of 1 d color ring projection patterns objects in the candidate locations are then verified by the pixel to pixel template matching to make the pixel based matching invariant to rotation a color feature is used as pixel density and the axis of least second moment is employed to estimate the rotational angle of a colored pattern the proposed method has shown promising result based on experiments on a variety of natural and industrial images

enumerative sequences of leaves and nodes in rational trees we prove that any n rational sequence s sn n 1 of nonnegative integers satisfying the kraft strict inequality n 1snk n 1 is the enumerative sequence of leaves by height of a rational k ary tree we give an efficient algorithm to get a k ary rational tree particular cases of this result had been previously proven we give some partial results in the case of equality especially we study the similar problem of characterizing the enumerative sequences of nodes of k ary rational trees and solve this question when the sequence has a primitive linear representation

effective computation of the multivariable alexander polinomial of lorenz links given two different representations of a lorenz link we compare how they affect the computation of the multivariable alexander polynomial we also compare the alexander polynomial with the trip number and genus our experimental results lead us to conjecture that for lorenz knots the alexander polynomial is an equivalent invariant to the pair trip number genus finally we give a counterexample in the case of lorenz links

asymptotic formulae for lorenz and horseshoe knots we derive various asymptotic formulae for the numbers of closed orbits in the lorenz and horseshoe templates with given knot invariants for example braid index and genus we indicate how these estimates can be applied to more complicated flows by giving a bound for the genus of knotted periodic orbits in the figure of eight template

a performance evaluation of local descriptors in this paper we compare the performance of descriptors computed for local interest regions as for example extracted by the harris affine detector mikolajczyk k and schmid c 2004 many different descriptors have been proposed in the literature it is unclear which descriptors are more appropriate and how their performance depends on the interest region detector the descriptors should be distinctive and at the same time robust to changes in viewing conditions as well as to errors of the detector our evaluation uses as criterion recall with respect to precision and is carried out for different image transformations we compare shape context belongie s et al april 2002 steerable filters freeman w and adelson e setp 1991 pca sift ke y and sukthankar r 2004 differential invariants koenderink j and van doorn a 1987 spin images lazebnik s et al 2003 sift lowe d g 1999 complex filters schaffalitzky f and zisserman a 2002 moment invariants van gool l et al 1996 and cross correlation for different types of interest regions we also propose an extension of the sift descriptor and show that it outperforms the original method furthermore we observe that the ranking of the descriptors is mostly independent of the interest region detector and that the sift based descriptors perform best moments and steerable filters show the best performance among the low dimensional descriptors

the skein polynomial of periodic knots a knot k in s 3 is said to have period r if there exists an orientation preserving homeomorphism f of period r on s 3 which preserves k with f i x f s 1 and with fix f c k o by the positive solution of smith conjecture f is a rotation of s 3 with unknotted axis fix f the purpose of this paper is to present the unexpectedly strong regularity of the skein polynomial 12 for periodic knots which was partially conjectured by traczyk here the skein polynomial pl v z of an oriented link l is uniquely defined by po v z 1 and

fast and accurate method for high order zernike moments computation abstract zernike moments zms which belong to a class of circularly orthogonal rotation invariant moments are very useful image descriptors for various image processing applications they however suffer from various errors such as discretization error geometric error and numerical integration error in this paper we present an approach for their computation which reduces geometric error and numerical integration error geometric error is reduced by performing exact mapping of a square grid domain into a circular grid by using both the square and arc grids numerical integration error is reduced by resorting to gaussian numerical integration over the square and arc grids it is observed that the contribution of numerical integration error to the total error is much more than the geometric error the effect of these two errors is more significant in small images as compared to large images the use of numerical integration however enhances the time complexity this problem is resolved by using recursive algorithms for the computation of radial polynomials 8 way symmetry of radial polynomials and 8 way symmetry anti symmetry of angular part of the kernel function exhaustive experiments on synthetic and real images are performed to demonstrate the effectiveness of the proposed approach

on the crossing number of positive knots and braids and braid index criteria of jones and morton williams franks we give examples of knots with some unusual properties of the crossing number of positive diagrams or strand number of positive braid representations in particular we show that positive braid knots may not have positive minimal strand number braid representations giving a counterpart to results of franks williams and murasugi other examples answer partially questions of cromwell on homogeneous and of adams on almost alternating knots we give a counterexample to and a corrected version of a theorem of jones on the alexander polynomial of 4 braid knots we also give an example of a knot on which all previously applied braid index criteria fail to estimate sharply from below the braid index a relation between generalizations of such examples and a conjecture of jones that a minimal braid representation has unique writhe is discussed

bennequin s inequality and the positivity of the signature we use an algorithm for special diagrams to prove a bennequin type inequality tor the signature of an arbitrary link diagram related to its murasugi sum decomposition we apply this inequality to show that the signature of a non trivial positive 3 braid knot is greater than its genus and that the signature of a positive braid link is minorated by an increasing function of its negated euler characteristic the latter property is conjectured to extend to positive links

efficient legendre moment computation for grey level images legendre orthogonal moments have been widely used in the field of image analysis because their computation by a direct method is very time expensive recent efforts have been devoted to the reduction of computational complexity nevertheless the existing algorithms are mainly focused on binary images we propose here a new fast method for computing the legendre moments which is not only suitable for binary images but also for grey level images we first establish a recurrence formula of one dimensional 1d legendre moments by using the recursive property of legendre polynomials as a result the 1d legendre moments of order p l p l p 0 can be expressed as a linear combination of l p 1 1 and l p 2 0 based on this relationship the 1d legendre moments l p 0 can thus be obtained from the arrays of l 1 a and l 0 a where a is an integer number less than p to further decrease the computation complexity an algorithm in which no multiplication is required is used to compute these quantities the method is then extended to the calculation of the two dimensional legendre moments l p q we show that the proposed method is more efficient than the direct method

robust and efficient image alignment based on relative gradient matching in this paper we present a robust image alignment algorithm based on matching of relative gradient maps this algorithm consists of two stages namely a learning based approximate pattern search and an iterative energy minimization procedure for matching relative image gradient the first stage finds some candidate poses of the pattern from the image through a fast nearest neighbor search of the best match of the relative gradient features computed from training database of feature vectors which are obtained from the synthesis of the geometrically transformed template image with the transformation parameters uniformly sampled from a given transformation parameter space subsequently the candidate poses are further verified and refined by matching the relative gradient images through an iterative energy minimization procedure this approach based on the matching of relative gradients is robust against nonuniform illumination variations experimental results on both simulated and real images are shown to demonstrate superior efficiency and robustness of the proposed algorithm over the conventional normalized correlation method

knotted periodic orbits in suspensions of annulus maps we consider a class of suspensions of diffeomorphisms of the annulus as flows in the orientable 3 manifold t 2 x i using results of birman williams topology 22 47 82 1983 contemp math 20 1 60 1983 we construct a knotholder or template that carries the set of periodic orbits of the flow we define rotation numbers and show that any orbit of period q and rotation number p q can be arranged as a positive braid on p strands this yields existence and uniqueness results for families of resonant torus knots p braids that are p q torus knots of period q p which correspond to order preserving birkhoff periodic orbits of the diffeomorphism we show that all other q periodic p braids have higher genus and we establish bounds on the genera of such knots we obtain existence and uniqueness results for a number of other non resonant torus knots including non order preserving q s q torus knots of rotation number 1

fast computation of legendre and zernike moments this paper presents recursive algorithms for fast computation of legendre and zernike moments of a grey level image intensity distribution for a binary image a contour integration method is developed for the evaluation of legendre moments using only the boundary information a method for recursive calculation of zernike polynomial coefficients is also given a square to circular image transformation scheme is introduced to minimize the computation involved in zernike moment functions the recursive formulae can also be used in inverse moment transforms to reconstruct the original image from moments the mathematical framework of the algorithms is given in detail and illustrated with binary and grey level images

rotation discriminating template matching based on fourier coefficients of radial projections with robustness to scaling and partial occlusion we consider brightness contrast invariant and rotation discriminating template matching that searches an image to analyze a for a query image q we propose to use the complex coefficients of the discrete fourier transform of the radial projections to compute new rotation invariant local features these coefficients can be efficiently obtained via fft we classify templates in stable and unstable ones and argue that any local feature based template matching may fail to find unstable templates we extract several stable sub templates of q and find them in a by comparing the features the matchings of the sub templates are combined using the hough transform as the features of a are computed only once the algorithm can find quickly many different sub templates in a and it is suitable for finding many query images in a multi scale searching and partial occlusion robust template matching

using orientation codes for rotation invariant template matching abstract a new method for rotation invariant template matching in gray scale images is proposed it is based on the utilization of gradient information in the form of orientation codes as the feature for approximating the rotation angle as well as for matching orientation codes based matching is robust for searching objects in cluttered environments even in the cases of illumination fluctuations resulting from shadowing or highlighting etc we use a two stage framework for realizing the rotation invariant template matching in the first stage histograms of orientation codes are employed for approximating the rotation angle of the object and then in the second stage matching is performed by rotating the object template by the estimated angle matching in the second stage is performed only for the positions which have higher similarity results in the first stage thereby pruning out insignificant locations to speed up the search experiments with real world scenes demonstrate the rotation and brightness invariance of the proposed method for performing object search

braids transversal links and the khovanov rozansky theory we establish some inequalities for the khovanov rozansky cohomologies of braids these give new upper bounds of the self linking numbers of transversal links in standard contact s 3 which are sharper than the well known bound given by the homfly polynomial we also introduce a sequence of transversal link invariants and discuss some of their properties

spiral aggregation map splam this paper describes a robust template matching algorithm undergoing rotation scaling translation rst variations via our proposed spiral aggregation map splam which is a novel image warping scheme it not only provides an efficient method for generating the desired projection profiles for matching it also enables us to determine the rotation angle and is invariant to scale changes compared to other model based methods the proposed spiral projection model spm provides the structural and statistical information about the template in a more general and easier to comprehend format the spm is a model based texture description scheme that enables the simultaneous representation for each value of projection profile the profile a set of parametric projection values functions by angular indexing is the aggregate from a group of spiral sampling pixels the experimental evaluation shows that the properties of the algorithm achieved very attractive results this paper describes a robust and fast template matching algorithm we introduced a novel image warping scheme called the spiral aggregation map splam this descriptor is capable of expressing the template with high order texture characteristics

dichromatic link invariants we investigate the skein theory of oriented dichromatic links in s3 we define a new chromatic skein invariant for a special class of dichromatic links this invariant generalizes both the two variable alexander polynomial and the twisted alexander polynomial alternatively one may view this new 1 2 invariant as an invariant of oriented monochromatic links in s x d and as such it is the exact analog of the twisted alexander polynomial we discuss basic properties of this new invariant and applications to link interchangeability for the full class of dichromatic links we show that there does not exist a chromatic skein invariant which is a mutual extension of both the two variable alexander polynomial and the twisted alexander polynomial

fast and numerically stable methods for the computation of zernike moments zernike moments zms are used in many image processing applications due to their superior performance over other moments however they suffer from high computation cost and numerical instability at high order of moments in the past many recursive methods have been developed to improve their speed performance and considerable success has been achieved the analysis of numerical stability has also gained momentum as it affects the accuracy of moments and their invariance property there are three recursive methods which are normally used in zms calculation prata s kintner s and q recursive methods the earlier studies have found the q recursive method outperforming the two other methods in this paper we modify prata s method and present a recursive relation which is proved to be faster than the q recursive method numerical instability is observed at high orders of moments with the q recursive method suffering from the underflow problem while the modified prata s method suffering from finite precision error the modified kintner s method is the least susceptible to these errors keeping in view the better numerical stability we further make the modified kintner s method marginally faster than the q recursive method we recommend the modified prata s method for low orders 90 and kintner s fast method for high orders 90 of zms

accurate calculation of image moments image moments have been extensively used as feature descriptors however the quantization error introduced in discrete signals presents problems especially when dealing with small size images this results in the fact that the invariant properties of moments are compromised in this paper we present a technique suitable for the calculation of moments from a continuous signal derived by piecewise polynomial interpolation of the corresponding discrete one the computed moments exhibit significantly increased accuracy while requiring trivial computational effort zernike moments are then computed using the proposed scheme and are shown to display increased stability to geometrical transformations

on image analysis by the methods of moments various types of moments have been used to recognize image patterns in a number of applications a number of moments are evaluated and some fundamental questions are addressed such as image representation ability noise sensitivity and information redundancy moments considered include regular moments legendre moments zernike moments pseudo zernike moments rotational moments and complex moments properties of these moments are examined in detail and the interrelationships among them are discussed both theoretical and experimental results are presented

on cobordisms between knots braid index and the upsilon invariant we use ozsvath stipsicz and szabo s upsilon invariant to provide bounds on cobordisms between knots that contain full twists in particular we recover and generalize a classical consequence of the morton franks williams inequality for knots positive braids that contain a positive full twist realize the braid index of their closure we also establish that quasi positive braids that are sufficiently twisted realize the minimal braid index among all knots that are concordant to their closure finally we provide inductive formulas for the upsilon invariant of torus knots and compare it to the levine tristram signature profile

partial classification of lorenz knots syllable permutations of torus knots words abstract we define families of aperiodic words associated to lorenz knots that arise naturally as syllable permutations of symbolic words corresponding to torus knots an algorithm to construct symbolic words of satellite lorenz knots is defined we prove subject to the validity of a previous conjecture that lorenz knots coded by some of these families of words are hyperbolic by showing that they are neither satellites nor torus knots and making use of thurston s theorem infinite families of hyperbolic lorenz knots are generated in this way to our knowledge for the first time the techniques used can be generalized to study other families of lorenz knots

quasipositivity as an obstruction to sliceness for an oriented link l s 3 d 4 let s l be the greatest euler characteristic f of an oriented 2 manifold f without closed components smoothly embedded in d 4 with boundary l a knot k is slice if s k 1 realize d 4 in c 2 as z w z 2 w 2 1 it has been conjectured that if v is a nonsingular complex plane curve transverse to s 3 then s v s 3 v d 4 kronheimer and mrowka have proved this conjecture in the case that v d 4 is the milnor fiber of a singularity i explain how this seemingly special case implies both the general case and the slice bennequin inequality for braids as applications i show that various knots are not slice e g pretzel knots like p 3 5 7 all knots obtained from a positive trefoil o 2 3 by iterated untwisted positive doubling