Liverpool University Articles and Preprints
by Hugh Morton and coworkers, (1987 onwards)

This is a list of my papers from 1987, with links to copies of some of them. Shortly my earlier papers should be available in the Liverpool University Repository.
Papers in this area by my colleagues Victor Goryunov, Toby Hall and Jonathan Woolf can be found through their own Liverpool University webpages.

Previous coworkers

I include here a list of many of my former students and collaborators. Some of their papers and theses are also listed below.
Programs for some knot theory calculations can be found in the list of programs.


You may also enjoy a collection of material by Elisabetta Beltrami and Peter Cromwell on the Borromean Rings.


  1. H.R.Morton,
    `Seifert circles and knot polynomials'.
    Math. Proc. Camb. Philos. Soc. 99 (1986), 107-109.
    Pdf version

  2. H.R.Morton and H.B.Short,
    `The 2-variable polynomial of cable knots'.
    Math. Proc. Camb. Philos. Soc. 101 (1987), 267-278.
    Pdf version

  3. H.R.Morton,
    `Polynomials from braids'.
    In `Braids', ed. Joan S. Birman and Anatoly Libgober, Contemporary Mathematics 78, Amer. Math. Soc. (1988), 375-385.
    Pdf version
    This paper is an improved version of a 1985 preprint originally entitled `Closed-braid representatives for a link and its 2-variable polynomial'.

  4. H.R.Morton and P.Traczyk,
    `The Jones polynomial of satellite links around mutants'.
    In `Braids', ed. Joan S. Birman and Anatoly Libgober, Contemporary Mathematics 78, Amer. Math. Soc. (1988), 587-592.

  5. P.R.Cromwell,
    `Homogeneous links'.
    J. London Math. Soc. 39 (1989), 535-552.

  6. H.R.Morton and H.B.Short,
    `Calculating the 2-variable polynomial for knots presented as closed braids'.
    J. Algorithms, 11 (1990), 117-131.

  7. H.R.Morton and P.Traczyk,
    `Knots and algebras'.
    In `Contribuciones Matematicas en homenaje al profesor D. Antonio Plans Sanz de Bremond', ed. E. Martin-Peinador and A. Rodez Usan, University of Zaragoza, (1990), 201-220. Pdf version (60K).

  8. J.M.Montesinos and H.R.Morton,
    `Generating fibred knots and links from closed braids'.
    Proc. London Math. Soc. 62 (1991), 167-201.
    Pdf version

  9. P.R.Cromwell,
    `Some infinite families of satellite knots with given Alexander polynomial'. Mathematika 38 (1991), 156-169.

  10. P.R.Cromwell,
    `Lonely knots and tangles; identifying knots with no companions'.
    Mathematika 38 (1991), 334-347.

  11. H.R.Morton and P.M.Strickland,
    `Jones polynomial invariants for knots and satellites'.
    Math. Proc. Camb. Philos. Soc. 109 (1991), 83-103.
    Pdf version

  12. H.R.Morton and P.M.Strickland,
    `Satellites and surgery invariants'.
    Preprint version
    In `Knots 90', Walter de Gruyter (1992), 47-66.
    Pdf version

  13. P.R.Cromwell and H.R.Morton,
    `Positivity of knot polynomials on positive links'.
    J. Knot Theory Ramif. 1 (1992), 203-206.
    Pdf version

  14. H.R.Morton,
    `Invariants of links and 3-manifolds from skein theory and from quantum groups'.
    In `Topics in knot theory', the Proceedings of the NATO Summer Institute in Erzurum 1992, NATO ASI Series C 399, ed. M.Bozhüyük. Kluwer 1993, 107-156. pdf version (420K), Gzipped postscript version (240K)

  15. H.R.Morton,
    `Quantum invariants given by evaluation of knot polynomials',
    J. Knot Theory Ramif. 2 (1993), 195-209.
    pdf version

  16. P.R.Cromwell,
    `Positive braids are visually prime',
    Proc. London Math. Soc. 67 (1993), 384-424.

  17. E.A.Elrifai and H.R.Morton,
    `Algorithms for positive braids'.
    Quart. J. Math. Oxford (2), 45 (1994), 479-497.
    Gzipped PostScript version (180K)
    pdf version

  18. P.M.Melvin and H.R.Morton,
    `The coloured Jones function'.
    Commun. Math. Phys. 169 (1995), 501-520.
    Gzipped PostScript version (120K)
    pdf version

  19. H.R.Morton,
    `The coloured Jones function and Alexander polynomial for torus knots'.
    Math. Proc. Camb. Philos. Soc. 117 (1995), 129-136.
    Pdf version

  20. P.R.Cromwell,
    `Embedding knots and links in an open book I: Basic properties'.
    Topology and its Applications 64 (1995), 37-58.

  21. P.R.Cromwell and I.J.Nutt,
    `Embedding knots and links in an open book II: Bounds on arc index'.
    Math. Proc. Camb. Philos. Soc. 119 (1996), 309-319.

  22. H.R.Morton and P.R.Cromwell,
    `Distinguishing mutants by knot polynomials'.
    J. Knot Theory Ramif. 5 (1996), 225-238.
    Pdf version

  23. H.J.Ryder,
    `The congruence structure on racks'.
    Comm. in Algebra. 23 (1995), 4971-4989.

  24. H.J.Ryder,
    `An algebraic condition to determine whether a knot is prime'.
    Math. Proc. Camb. Philos. Soc. 120 (1996), 385-389.

  25. H.R.Morton and H.J.Ryder,
    `Mutants and SU(3)q invariants'.
    In `Geometry and Topology Monographs', Vol.1: The Epstein Birthday Schrift.(1998), 365-381. Gzipped PostScript version (105K)
    Pdf version or from the Geometry and Topology website.

  26. I.J.Nutt,
    `Embedding knots and links in an open book III: On the braid index of satellite links'.
    Math. Proc. Camb. Philos. Soc., 126, (1999), 77-98..

  27. I.J.Nutt,
    `Arc index and the Kauffman polynomial'.
    J. Knot Theory Ramif. 6 (1998), 61-77.

  28. I.J.Nutt,
    `Braid index of satellite links'
    PhD thesis, University of Liverpool 1995. Pdf version (780K, about 160 pages)

  29. P.R.Cromwell,
    `Arc presentations of knots and links'.
    In Knot theory, (the proceedings of the knot theory conference, Warsaw 1995)
    Banach Center Publications 42 (1998), 57-64.
    Gzipped PostScript version (52K)

  30. E.Beltrami and P.R.Cromwell,
    `Minimal arc-presentations of some non-alternating knots'.
    Topology and its Applications 81 (1997), 137-145.
    Gzipped PostScript version (70K)

  31. E.Beltrami and P.R.Cromwell,
    `A limitation on algorithms for constructing minimal arc-presentations from link diagrams'.
    Preprint 1996. In J. Knot Theory Ramif. 7 (1998), 415-423.
    Gzipped PostScript version (74K)

  32. H.R.Morton and E.Beltrami,
    `Arc index and the Kauffman polynomial'.
    Math. Proc. Camb. Philos. Soc. 123 (1998), 41-48. Gzipped PostScript version (91K)
    Pdf version

  33. A.K.Aiston,
    `Skein theoretic idempotents of Hecke algebras and quantum group invariants.'
    PhD thesis, University of Liverpool 1996. Pdf version (800K)

  34. H.R.Morton and A.K.Aiston,
    `Idempotents of Hecke algebras of type A'.
    Preprint 1996, updated Jan. 1997.
    In J. Knot Theory Ramif. 7 (1998), 463-487. Pdf version (145K).
    Also available from q-alg 9702017

  35. H.R.Morton and A.K.Aiston,
    `Young diagrams, the Homfly skein of the annulus and unitary invariants'.
    Preprint 1996. In Proceedings of Knots96 (Shin'ichi Suzuki, ed.), World Scientific, Singapore. (1997), 31-45. Pdf version

  36. A.K.Aiston
    ` Adams operators and knot decorations'.
    Preprint 1995, updated 1997, available from q-alg 9711015

  37. A.K.Aiston
    ` A skein theoretic proof of the hook formula for quantum dimension'.
    Preprint 1997, available from q-alg 9711019

  38. H.R.Morton,
    `The Burau matrix and Fiedler's invariant for a closed braid'.
    Preprint April 1997, revised November 1997.
    In Topology and its Applications 95 (1999), 251-256. Pdf version (50K).
    Also available from the preprint archive math.GT 9801034

  39. H.R.Morton,
    `The multivariable Alexander polynomial for a closed braid'.
    Preprint 1997, revised March 1998. Pdf version (50K)
    In `Low-dimensional topology', ed. Hanna Nencka, Contemporary Mathematics 233, Amer. Math. Soc. (1999), 167-172.
    Also available from the preprint archive math.GT 9803138.
    You may also try (at your peril) using the Maple procedure to make calculations. The comments in the file tell you what can be calculated, and how to do it.

  40. H.R.Morton and M.Rampichini,
    `Mutual braiding and the band presentation of braid groups'.
    Preprint June 1998, revised September 1998.
    In Knots in Hellas 98, Proceedings of the International Conference on Knot Theory and its Ramifications, ed. C.Gordon et al., World Scientific (2000), 335-346. Gzipped PostScript version (80K).
    Pdf version (80K).
    Also available from the preprint archive math.GT 9907017

  41. H.R.Morton and A.J.Wassermann,
    `A basis for the Birman-Wenzl algebra'.
    Preprint 1989, with foreword and references updated Jan. 2000.
    Pdf file (237K).
    A version now appears in ArXiv arXiv:1012.3116.

  42. H.R.Morton and S.G.Lukac,
    `The Homfly polynomial of the decorated Hopf link'.
    Preprint July 2000, revised July 2001.
    In J. Knot Theory Ramif. 12 (2003), 395-416.
    Gzipped PostScript version (120K).
    Also available from the preprint archive math.GT 0108011

  43. S.G.Lukac,
    `Idempotents of the Hecke algebra become Schur functions in the skein of the annulus'.
    Preprint July 2000, revised October 2001.
    In Math. Proc. Cam. Philos. Soc., 138 (2005), 79-96. Pdf version
    Also available from the preprint archive math.GT 0110119

  44. Yongju Bae and H.R.Morton,
    `The spread and extreme terms of Jones polynomials'.
    Preprint, 2000, revised October 2001 and May 2002.
    In J. Knot Theory Ramif. 12 (2003), 359-373.
    Gzipped PostScript version (120K)
    Also available in the October 2001 version from the preprint archive math.GT 0012089

  45. H.R.Morton,
    `Skein theory and the Murphy operators'.
    Preprint, February 2001.
    In J. Knot Theory Ramif. 11 (2002), 475-492.
    Gzipped PostScript version. (112K)
    Also available from the preprint archive math.GT 0102098.

  46. S.Lukac,
    `Homfly skeins and the Hopf link'.
    PhD thesis, University of Liverpool 2001. Pdf version

  47. H.R.Morton and R.J.Hadji,
    ` Homfly polynomials of generalized Hopf links'.
    Preprint, June 2001.
    In Algebraic and Geometric Topology, 2 (2002), 11-32. Gzipped PostScript version. (132K)
    Also available from the preprint archive math.GT 0106207.

  48. H.R.Morton,
    `Power sums and Homfly skein theory'.
    Preprint, October 2001.
    In Geometry and Topology Monographs, 4 (2002) 'Invariants of knots and 3-manifolds (Kyoto 2001)' 235--244. Gzipped PostScript version. (95K)
    Also available from the preprint archive math.GT 0111101.

  49. H.R.Morton and R.J.Hadji,
    `Conjugacy for positive permutation braids'.
    Preprint, September 2003, updated December 2003.
    Pdf version. (70K)
    In Fundamentae Mathematicae 188 (2005), 155-166.
    Also available from the preprint archive math.GT 0312209.

  50. R.J.Hadji,
    'Homfly skein theory of reversed string satellites.'
    PhD thesis, University of Liverpool 2003. Pdf version

  51. R.J.Hadji and H.R.Morton,
    `A basis for the full Homfly skein of the annulus'.
    Preprint July 2004.
    Gzipped PostScript version. (120K)
    In Math. Proc. Camb. Philos. Soc., 141 (2006), 81-100.
    Pdf version
    Also available from the preprint archive math.GT 0408078.

  52. H.R.Morton,
    `The Alexander polynomial of a torus knot with twists'.
    Preprint December 2005, revised July 2006.
    In J. Knot Theory Ramif. , 15 (2006), 1037-1047.
    Gzipped PostScript version. (200K)
    Pdf version

  53. H.R.Morton,
    `Integrality of Homfly 1-tangle invariants'.
    Preprint June 2006.
    Gzipped PostScript version. (170K)
    In Algebraic and Geometric Topology, 7 (2007), 327-338.
    Also available from the preprint archive math.GT 0606336.

  54. H.R.Morton,
    `Mutant knots with symmetry'.
    Preprint May 2007.
    Pdf file. (200K)
    In Math. Proc. Camb. Philos. Soc., 146 (2009), 95-107.
    Also available from the preprint archive arXiv:0705.1321.

  55. H.R.Morton and P.M.G.Manchon,
    `Geometrical relations and plethysms in the Homfly skein of the annulus'.
    Preprint July 2007.
    Pdf file. (360K)
    In J. London Math. Soc. 78 (2008), 305-328.
    Also available from the preprint archive arXiv:0707.2851.

  56. H.R.Morton and N.Ryder,
    `Invariants of genus 2 mutants'.
    Preprint August 2007.
    Pdf file. (188K)
    In J. Knot Theory Ramif., 18 (2009), 1423-38.
    Also available from the preprint archive arXiv:0708.0514.

  57. H.R.Morton and S.Grishanov,
    'Doubly-periodic textile structures'.
    Preprint June 2008.
    Pdf file. (297K)
    In J. Knot Theory Ramif. 18 (2009), 1597-1622.
    Also available from the preprint archive arXiv:0806.2824.

  58. H.R.Morton and N.Ryder,
    `Relations between Kauffman and Homfly satellite invariants'.
    Preprint December 2008.
    Pdf file. (188K)
    In Math. Proc. Camb. Philos. Soc. 149 (2010), 105-114.
    Also available from the preprint archive arXiv:0902.1339.

  59. N.Ryder,
    'Skein based invariants and the Kauffman polynomial.'
    PhD thesis, University of Liverpool 2008. Pdf file (850K)

  60. H.R.Morton,
    'Mutant knots'.
    A survey of results about mutants, and their shared invariants.
    In New Ideas in Low-dimensional Topology ,
    World Scientific Series Knots and Everything 56, (2015), 379-412,
    eds. Louis.H.Kauffman and V.O.Manturov.
    Preprint February 2013, pdf file. (250K)

  61. S.Garoufalidis, H.R.Morton and T.Vuong,
    'The sl3 colored Jones polynomial of the trefoil'.
    In Proc. Amer. Math. Soc. 141 (2013), 2209-2220.
    Also available from the preprint archive arXiv:1010.3147.

  62. H.R.Morton and P.Samuelson,
    'The Homflypt skein algebra of the torus and the elliptic Hall algebra'
    To appear in Duke Mathematical Journal , Pdf file.
    Also available from the preprint archive arXiv:1410.0859.