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.

H.R.Morton,
`Seifert circles and knot polynomials'.
Math. Proc. Camb. Philos. Soc. 99 (1986), 107109.
Pdf version

H.R.Morton and H.B.Short,
`The 2variable polynomial of cable knots'.
Math. Proc. Camb. Philos. Soc. 101 (1987), 267278.
Pdf version

H.R.Morton,
`Polynomials from braids'.
In `Braids', ed. Joan S. Birman and Anatoly Libgober,
Contemporary Mathematics 78, Amer. Math. Soc. (1988), 375385.
Pdf version
This paper is an improved version of a 1985 preprint originally entitled `Closedbraid representatives for a link and its 2variable polynomial'.

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), 587592.

P.R.Cromwell,
`Homogeneous links'.
J. London Math. Soc. 39 (1989), 535552.

H.R.Morton and H.B.Short,
`Calculating the 2variable polynomial for knots presented as closed braids'.
J. Algorithms, 11 (1990), 117131.

H.R.Morton and P.Traczyk,
`Knots and algebras'.
In `Contribuciones Matematicas en homenaje al profesor D. Antonio
Plans Sanz de Bremond', ed. E. MartinPeinador and A. Rodez Usan,
University of Zaragoza, (1990), 201220.
Pdf version (60K).

J.M.Montesinos and H.R.Morton,
`Generating fibred knots and links from closed braids'.
Proc. London Math. Soc. 62 (1991), 167201.
Pdf version

P.R.Cromwell,
`Some infinite families of satellite knots with given Alexander polynomial'.
Mathematika 38 (1991), 156169.

P.R.Cromwell,
`Lonely knots and tangles; identifying knots with no companions'.
Mathematika 38 (1991), 334347.

H.R.Morton and P.M.Strickland,
`Jones polynomial invariants for knots and satellites'.
Math. Proc. Camb. Philos. Soc. 109 (1991), 83103.
Pdf version

H.R.Morton and P.M.Strickland,
`Satellites and surgery invariants'.
Preprint version
In `Knots 90', Walter de Gruyter (1992), 4766.
Pdf version

P.R.Cromwell and H.R.Morton,
`Positivity of knot polynomials on positive links'.
J. Knot Theory Ramif. 1 (1992), 203206.
Pdf version

H.R.Morton,
`Invariants of links and 3manifolds 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, 107156.
pdf version (420K),
Gzipped postscript version (240K)

H.R.Morton,
`Quantum invariants given by evaluation of knot polynomials',
J. Knot Theory Ramif. 2 (1993), 195209.
pdf version

P.R.Cromwell,
`Positive braids are visually prime',
Proc. London Math. Soc. 67 (1993), 384424.

E.A.Elrifai and H.R.Morton,
`Algorithms for positive braids'.
Quart. J. Math. Oxford (2), 45 (1994), 479497.
Gzipped PostScript version (180K)
pdf version

P.M.Melvin and H.R.Morton,
`The coloured Jones function'.
Commun. Math. Phys. 169 (1995), 501520.
Gzipped PostScript version (120K)
pdf version

H.R.Morton,
`The coloured Jones function and Alexander polynomial for torus knots'.
Math. Proc. Camb. Philos. Soc. 117 (1995), 129136.
Pdf version

P.R.Cromwell,
`Embedding knots and links in an open book I: Basic properties'.
Topology and its Applications 64 (1995), 3758.

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), 309319.

H.R.Morton and P.R.Cromwell,
`Distinguishing mutants by knot polynomials'.
J. Knot Theory Ramif. 5 (1996), 225238.
Pdf version

H.J.Ryder,
`The congruence structure on racks'.
Comm. in Algebra. 23 (1995), 49714989.

H.J.Ryder,
`An algebraic condition to determine whether a knot is prime'.
Math. Proc. Camb. Philos. Soc. 120 (1996), 385389.

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), 365381.
Gzipped PostScript version (105K)
Pdf version
or from the Geometry and Topology
website.

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), 7798..

I.J.Nutt,
`Arc index and the Kauffman polynomial'.
J. Knot Theory Ramif. 6 (1998), 6177.

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

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), 5764.
Gzipped PostScript version (52K)

E.Beltrami and P.R.Cromwell,
`Minimal arcpresentations of some nonalternating knots'.
Topology and its Applications
81 (1997), 137145.
Gzipped PostScript version (70K)

E.Beltrami and P.R.Cromwell,
`A limitation on algorithms for constructing minimal arcpresentations
from link diagrams'.
Preprint 1996. In J. Knot Theory Ramif. 7 (1998), 415423.
Gzipped PostScript version (74K)

H.R.Morton and E.Beltrami,
`Arc index and the Kauffman polynomial'.
Math. Proc. Camb. Philos. Soc. 123 (1998), 4148.
Gzipped PostScript version (91K)
Pdf version

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

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), 463487.
Pdf version (145K).
Also available from qalg 9702017

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), 3145.
Pdf version

A.K.Aiston
` Adams operators and knot decorations'.
Preprint 1995, updated 1997, available
from qalg 9711015
 A.K.Aiston
` A skein theoretic proof of the hook formula for
quantum dimension'.
Preprint 1997, available
from qalg 9711019

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), 251256.
Pdf version (50K).
Also available from the preprint archive
math.GT 9801034

H.R.Morton,
`The multivariable Alexander polynomial for a closed braid'.
Preprint 1997, revised March 1998.
Pdf version (50K)
In `Lowdimensional topology', ed. Hanna Nencka,
Contemporary Mathematics 233, Amer. Math. Soc. (1999), 167172.
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.

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), 335346.
Gzipped PostScript version (80K).
Pdf version (80K).
Also available from the preprint archive
math.GT 9907017

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

R.J.Hadji,
'Knots, tangles and algebras'.
MSc thesis, University of Liverpool 1999.
Pdf version

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), 395416.
Gzipped PostScript version (120K).
Also available from the preprint archive
math.GT 0108011

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), 7996.
Pdf version
Also available from the preprint archive
math.GT 0110119

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), 359373.
Gzipped PostScript version (120K)
Also available in the October 2001 version from the preprint archive
math.GT 0012089

H.R.Morton,
`Skein theory and the Murphy operators'.
Preprint, February 2001.
In J. Knot Theory Ramif. 11 (2002), 475492.
Gzipped PostScript version.
(112K)
Also available from the preprint archive
math.GT 0102098.

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

H.R.Morton and R.J.Hadji,
` Homfly polynomials of generalized Hopf links'.
Preprint, June 2001.
In Algebraic and Geometric Topology, 2 (2002), 1132.
Gzipped PostScript version. (132K)
Also available from the preprint archive
math.GT 0106207.

H.R.Morton,
`Power sums and Homfly skein theory'.
Preprint, October 2001.
In Geometry and Topology Monographs, 4 (2002)
'Invariants of knots and 3manifolds (Kyoto 2001)'
235244.
Gzipped PostScript version. (95K)
Also available from the preprint archive
math.GT 0111101.

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), 155166.
Also available from the preprint archive
math.GT 0312209.

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

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), 81100.
Pdf version
Also available from the preprint archive
math.GT 0408078.

H.R.Morton,
`Murphy operators in knot theory'.
Preprint 2005.
In `Differential Geometry and Physics',
Proceedings of the 23rd International Conference
on Differential Geometric Methods in Theoretical Physics,
Nankai Tracts in Mathematics vol 10, eds. MoLin Ge and Weiping Zhang.
World Scientific 2006, 359366.
pdf version

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), 10371047.
Gzipped PostScript version. (200K)
Pdf version

H.R.Morton,
`Integrality of Homfly 1tangle invariants'.
Preprint June 2006.
Gzipped PostScript version. (170K)
In Algebraic and Geometric Topology, 7 (2007), 327338.
Also available from the preprint archive
math.GT 0606336.

H.R.Morton,
`Mutant knots with symmetry'.
Preprint May 2007.
Pdf file. (200K)
In Math. Proc. Camb. Philos. Soc., 146 (2009), 95107.
Also available from the preprint archive
arXiv:0705.1321.

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), 305328.
Also available from the preprint archive
arXiv:0707.2851.

H.R.Morton and N.Ryder,
`Invariants of genus 2 mutants'.
Preprint August 2007.
Pdf file. (188K)
In J. Knot Theory Ramif., 18 (2009), 142338.
Also available from the preprint archive
arXiv:0708.0514.

H.R.Morton and S.Grishanov,
'Doublyperiodic textile structures'.
Preprint June 2008.
Pdf file. (297K)
In J. Knot Theory Ramif. 18 (2009), 15971622.
Also available from the preprint archive
arXiv:0806.2824.

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), 105114.
Also available from the preprint archive
arXiv:0902.1339.

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

H.R.Morton,
'Mutant knots'.
A survey of results about mutants, and their shared invariants.
In New Ideas in Lowdimensional Topology ,
World Scientific Series Knots and Everything 56, (2015), 379412,
eds. Louis.H.Kauffman and V.O.Manturov.
Preprint February 2013, pdf file. (250K)
 S.Garoufalidis, H.R.Morton and T.Vuong,
'The sl3 colored Jones polynomial of the trefoil'.
In Proc. Amer. Math. Soc. 141 (2013), 22092220.
Also available from the preprint archive
arXiv:1010.3147.
 H.R.Morton and P.Samuelson,
'The Homflypt skein algebra of the torus and the elliptic Hall algebra'
In Duke Mathematical Journal 166 (2017), 801854.
Pdf file.
Also available from the preprint archive arXiv:1410.0859.
 H.R.Morton,
'Satellites and quantum invariants'
An account of the interrelation between Homfly and Kauffman invariants and quantum invariants.
Pdf file.
Preprint March 2018. Prepared for inclusion in the Encyclopedia of Knot Theory.
 H.R.Morton and P.Samuelson,
'DAHAs and skein theory'
Pdf file.
In Commun. Math. Phys. 385 (2021), 16551693.
Also available from the preprint archive arXiv:1909.11247.
 H.R.Morton, A.Pokorny and P.Samuelson,
'The Kauffman skein algebra of the torus'
Pdf file.
Also available from the preprint archive arXiv:1909.11208.
 H.R.Morton,
'Tangles in affine Hecke algebras'
Pdf file.
To appear in J. Knot Theory Ramif.
Also available from the preprint archive arXiv:2005.12990.