Daniel S. Silver

Professor Emeritus

Department of Math and Statistics

Education

B.S. 1976
Trinity College, Hartford, Connecticut
Mathematics Major
Ph.D. in Mathematics, 1980
Yale University, New Haven, Connecticut

Research Interests

My research is in topology. I am especially interested in knots and links. Since 1995 I have been working with Susan G. Williams on applications of symbolic and algebraic dynamical systems to knot theory. I have also written articles and book reviews on the history of mathematics and science.

Publications

Mathematics Publications: (reprints and preprints are available on request.)

(with Susan Williams) Lehmer's Question, Graph Complexity Growth and Links, New York J. Math. (2021), in press.
A Colorful Approach to Knot Theory (or: “How Happy I Could Be With Ether”), 2020 Lewis-Parker Lecture, Alabama J. Math., in press.
(with Susan Williams) Group presentations for links in thickened surfaces, J. Knot Theory Ramifications 30 (2021), no. 3, 2150022, 19 pp.
(with Susan Williams) Persistent invariants of tangles, Chapter 14 of Encyclopedia of Knot Theory, (eds. C. Adams, E. Flapan, A. Henrich, L. Kaufman, L. Ludwig, S. Nelson), Routledge, 2020.
(with Susan Williams) Links in surfaces and Laplacian modules, J. Knot Theory Ramifications 29 (2020).
(with Lorenzo Traldi and Susan Williams) Goeritz and Seifert matrices from Dehn presentations, Osaka J. Math. 57 (2020), no. 3, 663–677.
(with Susan Williams) Tangles and links: A view with trees, Journal of Knot Theory and its Ramifications 27 (2018), no. 12, 1850061, 7 pp.
(with Stefan Friedl and Susan Williams) The Turaev and Thurston norms, Pacific Journal of Mathematics 284 (2016), 365-382.

(with Kalyn Lamey and Susan Williams) Vertex-colored graphs, bicycle spaces and Mahler measure, Journal of Knot Theory and its Ramifications 25 (2016), no. 6, 1650033, 22 pp.
(with Susan Williams) On the component number of links from plane graphs, J. Knot Theory Ramifications 24 (2015), no. 1, 1520002, 5 pp.
(with Stefan Friedl and Susan Williams) Splittings of knot groups, Math. Ann. 362 (2015), no. 1-2, 401 - 424.
(with J. Scott Carter and Susan Williams) Invariants of links in thickened surfaces , Algebraic and Geometric Topology, 14 (2013), 1377 -- 1394.
(with J. Scott Carter and Susan Williams) Three dimensions of knot coloring , American Mathematical Monthly, 121 (2014), 506--514.
(with Susan Williams) Virtual genus of satellite links , Journal of Knot Theory and its Ramifications, 22 (2013), 1350008 (4 pages).
(with Susan Williams) Twisted Alexander invariants of twisted links , Journal of Knot Theory and its Ramifications, 21 (2012), 1250118 (17 pages).
(with Susan Williams) On modules over Laurent polynomial rings , Journal of Knot Theory and its Ramifications, 21 (2012), 1250007--1250010.
(with Susan Williams) Alexander-Lin twisted polynomials, Journal of Knot Theory and its Ramifications, 20 (2011), 427--434.
(with Susan Williams) On a theorem of Burde and de Rham, Journal of Knot Theory and its Ramifications, 20 (2011), 713--720.
(with Wilbur Whitten) Knot group epimorphisms, II, preprint, 2009.
(with Jonathan Hillman and Susan Williams) On reciprocality of twisted Alexander polynomials, Algebraic and Geometric Topology, 10 (2010), 1017 -- 1026.
(with Susan Williams and Wilbur Whitten) Knot groups with many killers, Bulletin of the Australian Mathematical Society, 81 (2010), 507 -- 513.
(with Susan Williams) Dynamics of twisted Alexander invariants Topology and its Applications, 156 (2009), 2795--2811.
(with Susan Williams) Twisted Alexander polynomials and representation shifts, Bull. London Math. Soc., 41 (2009), 535--540.
(with Susan Williams) Nonfibered knots and representation shifts, Proceedings of Postnikov Memorial Conference, Banach Center Publications, 85 (2009), 101--107.
(with J. Scott Carter, Mohamed Elhamdadi, Masahico Saito and Susan Williams) Virtual knot invariants from group biquandles and their cocycles, Journal of Knot Theory and its Ramifications, 18 (2009), 957--972.
(with Mikami Hirasawa and Kunio Murasugi) When does a satellite knot fiber?, Hiroshima Journal of Mathematics, 38 (2008), 411--423.
(with Susan Williams) Lehmer's question, knots and surface dynamics, Mathematical Proceedings of the Cambridge Philosophical Society, 143 (2007), 649-661.
(with Susan Williams) Twisted Alexander polynomials detect the unknot, Algebraic and Geometric Topology, 6 (2006), 1893-1907.
(with Susan Williams) Crowell's derived group and twisted polynomials, Journal of Knot Theory and its Ramifications, 15 (2006), 1079-1094.
(with Alexander Stoimenow and Susan Williams) Euclidean Mahler measure and twisted links, Algebraic and Geometric Topology, 6 (2006), 581-602.
(with Susan Williams) Lifting representations of Z-groups, Israel Journal of Mathematics, 152 (2006), 313-331.
(with Wilbur Whitten) Knot group epimorphisms,Journal of Knot Theory and its Ramifications, 15 (2006), 153-166.

(with Wilbur Whitten) Hyperbolic covering knots, Algebraic and Geometric Topology 5 (2005), 1451-1469.
(with Susan Williams) An invariant of finite group actions of shifts of finite type, Ergodic Theory and Dynamical Systems, 25 (2005), 1985-1996.
(with Jozef Przytycki and Susan Williams) 3-Manifolds, tangles and persistent invariants, Mathematical Proceedings of the Cambridge Philosophical Society, 139 (2005), 291-306.
(with Susan Williams) Alexander groups of long virtual knots, Journal of Knot Theory and its Ramifications, 15 (2006), 43-52.
(with Susan Williams) An invariant of open virtual strings, Journal of Knot Theory and its Ramifications, 15 (2006), 143-152.
(with Susan Williams) Mahler measure of Alexander polynomials, J. London Math. Soc. (2) 69 (2004), 767-782.
(with Susan Williams) On a class of virtual knots with unit Jones polynomial, Knot Theory and its Ramifications 13 (2004), 367-371.
(with Susan Williams) Polynomial invariants of virtual links, Knot Theory and its Ramifications 12 (2003), 987-1000.
(with Susan Williams) Torsion numbers of augmented groups: with applications to knots and links , L'Enseignement Mathematiques 48 (2002), 317-343.
(with Susan Williams) Mahler measure, links and homology growth, Topology 41 (2002), 979-991.
(with Jenelle McAtee and Susan Williams) Coloring spatial graphs, J. Knot Theory and its Ramifications 10 (2001), 109-120.
(with Susan Williams) Alexander groups and virtual links, J. Knot Theory and its Ramifications 10 (2001), 151-160.
(with Susan Williams) A generalized Burau representation for string links, Pacific J. Math. 197 (2001), 241-255.
(with David Krebes and Susan Williams) Persistent invariants of tangles, J. Knot Theory and its Ramifications 9 (2000), 471-478.
(with Susan Williams) Virtual knot groups, Knots in Hellas '98 (World Scientific 2000), 440-451.
(with Susan Williams) Coloring link diagrams with a continuous palette, Topology 39 (2000), 1225-1237.
(with Istvan Kovacs and Susan Williams) Determinants of commuting-block matrices, Amer. Math. Monthly 106 (1999), 950-952.
(with Susan Williams) Virtual tangles and a theorem of Krebes, J. Knot Theory and its Ramifications 8 (1999), 941-945.
(with Susan Williams) Periodic links and augmented groups, Math. Proc. Cambridge Phil. Soc. 127 (1999), 217-236.
(with Susan Williams) Groups with uncountably many subgroups of finite index, Journal of Pure and Applied Algebra 140 (1999), 75-86.
(with Susan Williams) Knot invariants from symbolic dynamical systems, Trans. Amer. Math. Soc. 351 (1999), 3243-3265.
(with Susan Williams) Generalized n-colorings of links, Proceedings of the 1995 Conference in Knot Theory, Banach Center Publications 42 (1998), 373-386.
Free group automorphisms and knotted tori in the 4-sphere, Journal of Knot Theory and its Ramifications (1997) 6, 95-103.
(with Susan Williams) Knots, links and representation shifts, Proceedings of the 13th Annual Western Workshop on Geometric Topology (1996), 36-45.
(with Susan Williams) Augmented group systems and shifts of finite type, Israel Journal of Mathematics 95 (1996), 231-251.
HNN bases and high-dimensional knots, Proceedings of the American Mathematical Society (1996) 124, 1247-1252.
Nontorus knot groups are hyperhopfian, Bulletin of the London Mathematical Society, (1996) 28, 4-6.
Knot invariants from topological entropy, Topology and its Applications (1995) 61, 159-177.
Augmented group systems and n-knots, Mathematische Annalen (1993) 296, 585-593.
On the existence of minimal Seifert manifolds, Mathematical Proceedings of the Cambridge Philosophical Society (1993) 114, 103-109.
On knot-like groups and ribbon concordance, Journal of Pure and Applied Algebra (1992) 82, 99-105.
Growth rates of n-knots, Topology and its Applications (1991) 42, 217-230.
Delta-moves on links and Jones polynomial evaluations, Canadian Mathematical Bulletin (1991) 34, 393-400.
Examples of 3-knots with no minimal Seifert manifolds, Mathematical Proceedings of the Cambridge Philosophical Society (1991) 110, 417-420.
(with L. Richard Hitt) Ribbon knot families via Stallings' twists, Journal of the Australian Mathematical Society (1991) 50, 356-372.
An application of group endomorphism growth rates to knot theory, Proceedings of the Seventh Annual Western Workshop in Geometric Topology, Oregon State University (1990), 70-73.
Construction by isotopy, II, Transactions of the American Mathematical Society (1990) 317, 813-823.
On Aitchison's construction by isotopy, Transactions of the American Mathematical Society (1988) 305, 641-652.
(with Iain Aitchison) On fibring certain ribbon disc pairs, Transactions of the American Mathematical Society (1988) 305, 529-548.
(with Iain Aitchison) On a theorem of Yoshikawa, Transactions of the American Mathematical Society (1988) 305, 548-551 (appears as an appendix to previous paper).
Surgery in dimension four and non-compact 5-manifolds, Transactions of the American Mathematical Society (1985) 288, 43-50.
Finding stable-boundaries for open five-dimensional manifolds, American Journal of Mathematics (1984) 105, 1309-1324.
Embedding 2-spheres topologically in smooth 4-manifolds, Houston Journal of Mathematics (1983) 9, 119-124.

Reviews of Mathematics Books:

Review of "Do Not Erase: Mathematicians and their Chalkboards," by Jessica Wynne, Princeton University Press, in Notices of the American Mathematical Society, to appear.
Review of "Music by the Numbers: From Pythagoras to Schoenberg," by Eli Maor, Princeton University Press, in American Scientist, November/December 2018, 376-378.
Review of "A Mind at Play: How Claude Shannon Invented the Information Age," by Jimmy Soni and Rob Goodman, Simon and Schuster, in American Scientist, January/February 2018, 54-57.
Review of "Nathaniel Bowditch and the Power of Numbers: How a Nineteenth-Century Man of Business, Science, and the Sea Changed American Life," by Tamara Plakins Thornton, University of North Carolina Press, in American Scientist, November/December 2016, 374-378.
Review of "Creating Symmetry: The Artful Mathematics of Wallpaper Patterns," by Frank A. Farris, Princeton University Press, 2015, in American Scientist, September/October 2015, 360-362.
Review of "An Invitation to Mathematics: From Competitions to Research, " edited by Dierk Schleicher and Malte Lackmann, Springer-Verlag, Mathematical Intelligencer, Vol. 37, Winter 2015.
Review of "Duel at Dawn, " by Amir Alexander, Harvard University Press, in November 2010 issue of Notices of the American Mathematical Society.
Review of "Tools of American Mathematics Teaching, 1800-2000," by Peggy Aldrich Kidwell, Amy Ackerberg-Hastings, and David Lindsay Roberts, Johns Hopkins University Press, American Mathematical Monthly, Vol. 117, February 2010, 190-192.
Review of "Arthur Cayley: Mathematician Laureate of the Victorian Age, " by Tony Crilly, Johns Hopkins University Press, and "James Joseph Sylvester: Jewish Mathematician in a Victorian World, " by Karen Hunger Parshall, in American Scientist, November-December 2006.
Review of "Four Colors Suffice: How the Map Problem was Solved,", by Robin Wilson, Princeton University Press, in American Scientist, July-August 2003.
Review of "Knots," by G. Burde and H. Zieschang, de Gruyter Stud. Math., vol. 5, Walter de Gruyter, NY 2003 and "Algebraic invariants of links,", by J. Hillman, Series on Knots and Everything, Vol. 32, World Scientific Co., Pte. Ltd., Singapore 2002, in Bulletin of the American Mathematical Society, 41 (2003), 135-147.
Review of "Oxford Figures: 800 Years of the Mathematical Sciences," edited by John Fauvel, Raymond Flood and Robin Wilson, Oxford University Press, in Alabama Journal of Mathematics 26 (2002), 69-72.
Review of "James Joseph Sylvester: Life and Works in Letters," by Karen Hunger Parshall, Oxford University Press, in Alabama Journal of Mathematics (1999) 23, 49-50.

History and Philosophy of Mathematics:

Stop Me If You've Heard This Theorem Before, American Scientist, September/October 2020 (E-prints available. Please ask.)
The New Language of Mathematics, American Scientist, November/December 2017 (E-prints available. Please ask.)
Mathematical Induction and the Nature of British Miracles, American Scientist, September/October 2016 (E-prints available. Please ask.)
Seventy-five Years of Apology by G.H. Hardy, American Scientist, November/December 2015 (E-prints available. Please ask.) The article will appear in The Best Writing in Mathematics 2016, Princeton University Press.
Knots in the Nursery: (Cats) Cradle Song of James Clerk Maxwell , Notices of the American Mathematical Society, 61 (2014), 1186 -- 1194.
New introduction to "Riddles in Mathematics, " by Eugene P. Northrop, Dover Publications, 2014.
Slicing a Cone for Art and Science: Renaissance Artist Albrecht Dürer Searched for Beauty with Mathematics, American Scientist, September-October 2012. (E-reprints are available. Please ask me.) The article also appears in The Best Writing in Mathematics 2013, Princeton University Press.
New introduction to "Topology of 3-Manifolds and Related Topics, " edited by M.K. Fort, Jr., Dover Publications, 2010.
The Last Poem of James Clerk Maxwell , Notices of the American Mathematical Society, 55 (2008), 1266-1270.
Knot Theory's Odd Origins, American Scientist, 94, No. 2, 158-165. (E-reprints are available. Please ask me.)
Scottish physics and knot theory's odd origins, July, 2005 (Earlier manuscript that appeared in shorter, edited form as "Knot theory's odd origins.")
"Perhaps I Might Explain This": The Toys and Humor of James Clerk Maxwell, Textualities, Science and Intuition issue, a literary magazine Textualities

Resources on History of Knot Theory

A 5-Minute History of Knot Theory
Smokey pas de deux gif produced by artist at American Scientist
When Smokes Gets in Your Eyes
James Clerk Maxwell’s hand-drawn zoetrope illustration of three smoke rings (Copyright is owned by Cavendish Museum and reproduced here with their permission)
Why Knot? An elementary exposition on Knots and Fox colorings for undergraduates