Geometric Topology

Knot Theory

My principal interests here are

• Classification of high-dimensional simple knots.
• Factorisation of high-dimensional knots.
• Double-null-cobordism of knots.
• Branched cyclic covers of knots / cyclic group actions on spheres.
• Algebraic invariants of knots, especially those related to Blanchfield duality and the knot module.

Singularity Theory

My principal interest here is

• Isolated singularities of complex hypersurfaces and the Milnor fibre.

Preprints

• Prime simple knots in higher dimensions.
  with S.M.J. Wilson.
  
• Branched cyclic covers of knots.
  with S.M.J. Wilson.
  

Publications

• New invariants of simple knots.
  Journal of Knot Theory and its Ramifications 17 (2008) 337-350.
  with S.M.J. Wilson.
  
• S-equivalence of knots.
  Journal of Knot Theory and its Ramifications 13 (2004) 709-717.
• Sharp bounds on some classical knot invariants.
  with S.M.J. Wilson.
  Journal of Knot Theory and its Ramifications 12 (2003) pp. 805-817.
• Simple non-finite knots are not prime in higher dimensions.
  with S.M.J. Wilson.
  Journal of Knot Theory and its Ramifications 12 (2003) pp. 225-241.
• Knot modules and the Nakanishi index.
  with S.M.J. Wilson.
  Proceedings of the American Mathematical Society 131 (2003) pp. 655-663.
  
• Spinning and branched cyclic covers of knots.
  with S.M.J. Wilson.
  Royal Society of London Proceedings A 455 (1999) pp. 2235-2244.
  
• Simple 4-knots.
  with J.A. Hillman:
  Journal of Knot Theory and its Ramifications 7 (1998) pp. 907-923.
• Finiteness results for knots and singularities.
  with O. Saeki, S.M.J. Wilson:
  Kobe Journal of Mathematics 14, 2 (1997) pp. 197-206.
• Alexander ideals of classical knots.
  with S.M.J. Wilson:
  Publicacions Matemàtiques, 41 (1997) pp. 489-494.
• Algebraic invariants of simple 4-knots.
  with J.A. Hillman:
  Journal of Knot Theory and its Ramifications, 6 (1997) pp. 307-318.
• Cyclic group actions and knots,
  with S.M.J. Wilson:
  Royal Society of London Proceedings A, 436 (1992) pp. 527-535.
• Spherical algebraic knots increase with dimension,
  with J.H.M. Steenbrink:
  Archiv der Mathematik 57 (1991) pp. 518-520.
• Knots, groups and spinning,
  Glasgow Mathematical Journal 33 (1991) pp. 99-100.
• Finiteness theorems for conjugacy classes and branched cyclic covers of knots,
  with E. Bayer-Fluckiger and S.M.J. Wilson:
  Mathematische Zeitschrift, 201 (1989) pp. 485-493.
• Stable isometry structures and the factorization of Q-acyclic stable knots,
  with J.A. Hillman:
  Journal of the London Mathematical Society, 39 (1989) pp. 175-182.
• Hermitian forms in additive categories: finiteness results,
  with E. Bayer-Fluckiger and S.M.J. Wilson:
  Journal of Algebra, 123 (1989) pp. 336-350.
• Mutation of knots,
  Proceedings of the American Mathematical Society, 105 (1989) pp. 206-208.
• Seifert matrices and 6-knots,
  with J.A. Hillman:
  Transactions of the American Mathematical Society, 309 (1988) pp. 843-855.
• Decomposition of modules, forms and simple knots,
  with E. Bayer-Fluckiger and S.M.J. Wilson:
  Journal fur die reine und angewandte Mathematik, 375/376 (1987) pp. 167-183.
• Integer invariants of certain even-dimensional knots,
  Proceedings of the American Mathematical Society, 93 (1985) pp. 747-750.
• Doubly-null-cobordant simple even-dimensional knots,
  Proceedings of the Royal Society of Edinburgh, 96A (1984) pp. 163-174.
• Simple spun knots,
  Topology, 23 (1984) pp. 91-95.
• Some non-fibred 3-knots,
  Bulletin of the London Mathematical Society, 15 (1983) pp. 365-367.
• Spinning, factorisation of knots and cyclic group actions on spheres,
  Archiv der Mathematik 40 (1983) pp. 361-363.
• An algebraic classification of certain simple even-dimensional knots,
  Transactions of the American Mathematical Society, 276 (1983) pp. 1-53.
• A remarkable 3-knot,
  Bulletin of the London Mathematical Society, 14 (1982) pp. 397-398.
• The factorisation of knots,
  Low-dimensional Topology (Proceedings of the 1979 Bangor Conference),
  London Mathematical Society Lecture Notes, 48 (1982) pp. 71-80.
• with S.M.J. Wilson:
  Cyclic group actions on odd-dimensional spheres,
  Commentarii Mathematici Helvetici, 56 (1981) pp. 615-626.
• with E. Bayer and J.A. Hillman:
  The factorization of simple knots,
  Mathematical Proceedings of the Cambridge Philosophical Society, 90 (1981) pp. 495-506.
• Hermitian signatures and double-null-cobordism of knots,
  Journal of the London Mathematical Society (2), 23 (1981) pp. 563-572.
• An algebraic classification of certain simple knots,
  Archiv der Mathematik 35 (1980) pp. 391-393.
• Obstructions to embedding and isotopy in the metastable range,
  Mathematische Annalen 243 (1979) pp. 103-113.
• The Milnor signatures of compound knots,
  Proceedings of the American Mathematical Society, 76 (1979) pp. 157-160.
• Factorisation is not unique for 3-knots,
  Indiana University Mathematics Journal, 28 (1979) pp. 451-452.
• Signatures of knots and the free differential calculus,
  Quarterly Journal of Mathematics, 30 (1979) pp. 157-182.
• Attempting to classify knot modules and their Hermitian pairings,
  Knot Theory, Les Plans sur Bex, 1977,
  Springer Lecture Notes, 685 (1978) pp. 227-242.
• An algebraic classification of some even-dimensional knots,
  Topology 15 (1975) pp. 363-373.
• Cobordism of knots and Blanchfield duality,
  Journal of the London Mathematical Society (2), 10 (1975) pp. 406-408.
• Simple knots which are doubly-null-cobordant,
  Proceedings of the American Mathematical Society, 52 (1975) pp. 471-472.
• Blanchfield duality and simple knots,
  Transactions of the American Mathematical Society, 202 (1975) pp. 141-160.
• Presentations of n-knots,
  Transactions of the American Mathematical Society, 202 (1975) pp. 123-140.
• Non-invertible knots of codimension two, (PDF)
  Proceedings of the American Mathematical Society, 40 (1973) pp. 274-276.
• Classification of simple knots by Blanchfield duality,
  Bulletin of the American Mathematical Society, 79 (1973) pp. 952-955.
• A theorem on twisted spun knots,
  Bulletin of the London Mathematical Society, 4 (1972) pp. 47-48.
• Piecewise-linear critical levels and collapsing,
  with W.B.R. Lickorish:
  Transactions of the American Mathematical Society, 172 (1972) pp. 415-424.
• Regular neighbourhoods and mapping cylinders,
  Proceedings of the Cambridge Philosophical Society, 70 (1971) pp. 15-18.

Survey Articles

• Quadratic Forms in Knot Theory. Postscript or PDF
  Quadratic Forms and Their Applications, University College Dublin, 1999.
  American Mathematical Society Contemporary Mathematics 272 (2000) 135-154.

Talks

• Spinning and Branched Cyclic Covers of Knots. Postscript or PDF
• Quadratic Forms in Knot Theory. Postscript or PDF

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