Email: traldil at lafayette.edu
Department of Mathematics (emeritus)
Lafayette College
Easton, PA 18042 USA

Thanks to Ivan Dimitrov for this picture of commencement 2007

I grew up in New York City, a block from the American Museum of Natural History (really!). I went to Bronx Science for high school, and then Queens College. At Queens in 1973 I fell in love twice, once with mathematics — 88 of the 120 credits I took as an undergraduate were in math. I also met Sharon Richter, and we’ve been married since 1977. We have four wonderful children — Arthur, Matthew, Oliver and Rebecca — and when they’re home our house is full of music, chess and conversation. See some of our old family photos and Becky and Johan’s wedding pictures.

I got my Ph.D. from Yale in 1980, and I came to Lafayette right after graduation. My Ph.D. dissertation was concerned with the elementary ideals of knots and links in three dimensions. A knot is simply a circle imbedded in 3-space, and a link is an assembly of several imbedded circles. Their elementary ideals are invariants; that is, if one link can be transformed into another in space, without cutting, then they will have the same elementary ideals. Some dramatic knot-theoretic advances of the 1980s used combinatorics rather than algebraic topology to study knots and links, and after learning about them I studied several different kinds of combinatorial structures. I’m particularly interested in matroids, which generalize graphs and vector spaces, and clutters, which are used to describe network reliability problems. The most familiar examples of clutters consist of the sets of edges of a graph which form circuits or spanning trees. An important idea of network reliability analysis is to study partitions of the family that contains every set which contains at least one element of a clutter — for example, the family that contains every subset of the edge-set of a connected graph which contains the edge-set of at least one spanning tree — and I’ve studied these partitions using graph theory, iterative techniques and symbolic logic. In 2003 I learned about generalized dice games and the fascinating field of voting theory, which is full of counterintuitive results. More recently I’ve gone back to studying knot theory, and related ideas involving circuits in 4-regular graphs and linear algebra.

I taught lots of courses at Lafayette: precalculus, finite math, calculus, statistics, discrete math, a first-year seminar on paradoxes, linear algebra, geometry, graph theory, foundations (logic and set theory), complex calculus, abstract algebra I and II, real analysis I and II, topology and algebraic topology. I’ve also worked with many independent study students. Seven students wrote honors theses with me: Daniel L. King ’87 wrote “A structure theorem for semi-primitive Artinian rings,” Ashish Bhan ’91 wrote “A structure theorem for modules over pids,” Jennifer J. Mandelson ’97 wrote “Evariste Galois, the man and his mathematics,” Andrew Hahn ’98 wrote “An introduction to cardinality theory and infinity,” Alexandru O. Balan ’03 wrote “An enhanced approach to network reliability using Boolean algebra” and Yemai Liu ’15 wrote “A brief history of linear algebra and two important theorems.” Bidur Dahal ’12 wrote “A survey of Tutte polynomials and T-uniqueness” with Gary Gordon and me.

I’ve been glad to participate in the Research Experiences for Undergraduates program sponsored by the NSF and Lafayette College. Ryan R. Martin, Justin T. Moore, Jessica K. Benashski ’96 and I worked on a project in graph theory during the summer of 1994. Jonah Blasiak, Jennifer M. Rowe and OdedYacobi studied clutters with me during the summer of 2001, and so did Michael H. Dinitz, Jonah M. Gold and Thomas C. Sharkey during the summer of 2003. In the summer of 2005 Jeff Johannes, Karen McCready, Ken Ober and Matthias Youngs and I studied finite-type link invariants. Chelsey Cooley, Billy Ella, Michael Follett ’11, Eric Gilson and I learned about dice games as examples of the theory of voting in the summer of 2008.

I’ve also been glad to work with students supported by Lafayette’s EXCEL Scholars program. William R. Graveman ’99 worked on clutters with me in the summer of 1997. Alexandru O. Balan ’03 and I studied Boolean algorithms for network reliability during the summers of the years 2000 – 2002, and Rebecca Andersen ’06 and Jonathan Rowe ’06 continued that work during the summers of 2004 and 2005; you can download an algorithm we developed here. Brian Kronenthal ’07 and Michael Follett ’11 helped me study dice games in 2006, 2008 and 2009. Joe Dudek ’09 and Mike Follett ’11 worked with Louis Zulli and me in 2007, 2008 and 2009 to understand how certain invariants of knots and graphs are related to linear algebra over the two-element field.

Mathematical publications

Note: Please email me if you would like a copy of a paper which is not “clickable” below, or if clicking does not provide you with the final version of a published paper. (Some publishers prefer that authors not post published articles online, but allow distribution to individuals.)

Multivariate Alexander quandles, V. Constructing the medial quandle of a link

Circuit partitions and signed interlacement in 4-regular graphs

Binary matroids that classify forests, Algebra and Discrete Mathematics

Multivariate Alexander quandles, IV. The medial quandle of a link, Journal of Knot Theory and its Ramifications 31 (2022), article 2250077

Peripheral elements in reduced Alexander modules, Journal of Knot Theory and its Ramifications 31 (2022), article 2250058

Link mutations and Goeritz matrices, Osaka Journal of Mathematics 59 (2022), 881-904

A note on geometric duality in matroid theory and knot theory, Discrete Applied Mathematics 320 (2022), 184-190

(with R. Brijder) A characterization of circle graphs in terms of total unimodularity, European Journal of Combinatorics 102 (2022), 103455

Linking numbers, quandles and groups, Journal of Knot Theory and its Ramifications 30 (2021), article 2150048

(with D. A. Smith and W. Watkins) Planarity, duality and Laplacian congruence, Journal of Combinatorial Mathematics and Combinatorial Computing 113 (2020), 299–321

(with D. S. Silver and S. G. Williams) Goeritz and Seifert matrices from Dehn presentations, Osaka Journal of Mathematics 57 (2020), 663-677

(with R. Brijder) A characterization of circle graphs in terms of multimatroid representations, Electronic Journal of Combinatorics 27 (2020), #P1.25

Multivariate Alexander quandles, II. The involutory medial quandle of a link (corrected), Journal of Knot Theory and its Ramifications 29 (2020), article 2050093

Multivariate Alexander quandles, I. The module sequence of a link, Journal of Knot Theory and its Ramifications 29 (2020), article 2050009

Multivariate Alexander quandles, III. Sublinks, Journal of Knot Theory and its Ramifications 28 (2019), article 1950090

(with D. A. Smith and W. Watkins) A note on 2-isomorphisms and the signed Laplacian matrix of a graph, Linear Algebra and its Applications 563 (2019), 277-286

Multivariate Alexander colorings, Journal of Knot Theory and its Ramifications 27 (2018), article 1850076

(with D. A. Smith and W. Watkins) A note on Dehn colorings and invariant factors, Journal of Knot Theory and its Ramifications 27 (2018), article 1871003

(with D. A. Smith and W. Watkins) Duality and the signed Laplacian matrix of a graph, Linear Algebra and its Applications 548 (2018), 1-18

(with S. Backman and S. Hopkins) Fourientation activities and the Tutte polynomial, European Journal of Combinatorics 67 (2018), 40-60

Link colorings and the Goeritz matrix, Journal of Knot Theory and its Ramifications 26 (2017), article 1750045

Notes on a theorem of Naji, Discrete Mathematics 340 (2017), 3217-3234

(with R. Brijder) Isotropic matroids III: Connectivity, The Electronic Journal of Combinatorics 24 (2017), #P2.49

(with R. Brijder) Isotropic matroids II: Circle graphs, The Electronic Journal of Combinatorics 23 (2016), #P4.2

(with R. Brijder) Isotropic matroids I: Multimatroids and neighborhoods, The Electronic Journal of Combinatorics 23 (2016), #P4.1

Splitting cubic circle graphs, Discussiones Mathematicae Graph Theory 36 (2016), 723-741

The transition matroid of a 4-regular graph: an introduction, European Journal of Combinatorics 50 (2015), 180-207

Binary matroids and local complementation, European Journal of Combinatorics 45 (2015), 21-40

(with C. Cooley, W. Ella, M. Follett ’11 and E. Gilson) Tied dice. II. Some asymptotic results, Journal of Combinatorial Mathematics and Combinatorial Computing 90 (2014), 241-248

Interlacement in 4-regular graphs: a new approach using nonsymmetric matrices, Contributions to Discrete Mathematics 9 (2014), 85-97

(with R. Brijder and H. J. Hoogeboom) The adjacency matroid of a graph, The Electronic Journal of Combinatorics 20 (2013), #P27

On the interlace polynomials, Journal of Combinatorial Theory, Series B 103 (2013), 184-208

(with E. Gilson, C. Cooley, W. Ella and M. Follett ’11) The Efron dice voting system, Social Choice and Welfare 39 (2012), 931–959

On the linear algebra of local complementation, Linear Algebra and its Applications 436 (2012), 1072–1089

Binary nullity, Euler circuits and interlace polynomials, European Journal of Combinatorics 32 (2011), 944-950

A bracket polynomial for graphs, IV. Undirected Euler circuits, graph-links and multiply marked graphs, Journal of Knot Theory and its Ramifications 20 (2011), 1093-1128

A bracket polynomial for graphs, III. Vertex weights, Journal of Knot Theory and its Ramifications 20 (2011), 435-462

A bracket polynomial for graphs, II. Links, Euler circuits and marked graphs, Journal of Knot Theory and its Ramifications 19 (2010), 547-586

(with R. E. Andersen ’06, A. O. Balan ’03 and J. P. Rowe ’06) A note on independent-disjoint analysis of network reliability, Applied Mathematical and Computational Sciences 1 (2010), 61-66

(with M. H. Dinitz, J. M. Gold and T. C. Sharkey) Graphical representations of clutters, Ars Combinatoria 94 (2010), 303-320

Weighted interlace polynomials, Combinatorics, Probability and Computing 19 (2010), 133-157

(with C. Anderson, J. Cutler and A. J. Radcliffe) On the interlace polynomials of forests, Discrete Mathematics 310 (2010), 31-36

(with L. Zulli) A bracket polynomial for graphs, I, Journal of Knot Theory and its Ramifications 18 (2009), 1681-1709

(with L. Zulli) Topological realizations of ortho-projection graphs, Topology and its Applications 156 (2009), 2515-2526

(with B. G. Kronenthal ’07) Tied dice, Journal of Combinatorial Mathematics and Combinatorial Computing 68 (2009), 85-96

(with B. Shtylla ’05 and L. Zulli) On the realization of double occurrence words, Discrete Mathematics 309 (2009), 1769-1773

Generalized dice: Many questions and a few answers, Graph Theory Notes of New York LIII (2007), 39-42

A conjecture about sums of disjoint products, Journal of Combinatorial Mathematics and Combinatorial Computing 60 (2007), 3-6

[Click here for symmetric SDP forms of the disjunction of p < 19 variables, or here for p = 19, or here for p = 23.]

(with J. A. Ellis-Monaghan) Parametrized Tutte polynomials of graphs and matroids, Combinatorics, Probability and Computing 15 (2006), 835-854

Dice games and Arrow’s theorem, Bulletin of the Institute of Combinatorics and its Applications 47 (2006), 19-22

On the colored Tutte polynomial of a graph of bounded treewidth, Discrete Applied Mathematics 154 (2006), 1032-1036

Non-minimal sums of disjoint products, Reliability Engineering and System Safety 91 (2006), 533-538

Dice graphs, Congressus Numerantium 172 (2005), 177-191

(with J. Blasiak, J. Rowe and O. Yacobi) Several definitions of matroids, Ars Combinatoria 77 (2005), 33-44

The prevalence of “paradoxical” dice, Bulletin of the Institute of Combinatorics and its Applications 45 (2005), 70-76

Generalized activities and reliability, Journal of Combinatorial Mathematics and Combinatorial Computing 52 (2005), 17-32

Parallel connections and coloured Tutte polynomials, Discrete Mathematics 290 (2005), 291-299

A subset expansion of the coloured Tutte polynomial, Combinatorics, Probability and Computing 13 (2004), 269-275

Rearrangements of generalized shellings, Congressus Numerantium 163 (2003), 27-32 unpublished early version

(with A. O. Balan ’03) Preprocessing minpaths for sum of disjoint products, Institute of Electrical and Electronics Engineers Transactions on Reliability 52 (2003), 289-295

Clutters and circuits, III, Algebra Universalis 49 (2003), 201-209

A note on delta-wye-delta reductions of plane graphs, Congressus Numerantium 158 (2002), 213-220

Visualizing clutters, Congressus Numerantium 154 (2002), 13-20

Chain polynomials and Tutte polynomials, Discrete Mathematics 248 (2002), 279-282

Commentary on: reliability polynomials and link importance in networks, Institute of Electrical and Electronics Engineers Transactions on Reliability 49 (2000), 322

A note on a theorem of Wu, Kobe Journal of Mathematics 17 (2000), 27-28

Series and parallel reductions for the Tutte polynomial, Discrete Mathematics 220 (2000), 291-297

A note on reliability and expected value, Congressus Numerantium 133 (1998), 95-99

Two universal reliability problems, Congressus Numerantium 132 (1998), 199-204

Clutters and circuits. II, Advances in Applied Mathematics 21 (1998), 437-456

Clutters and circuits, Advances in Applied Mathematics 18 (1997), 220-236

(with J. Rodriguez) (K,j)-domination and (K,j)-reliability, Networks 30 (1997), 293-306

On the Arf invariant of a purely proper link, Journal of Knot Theory and its Ramifications 5 (1996), 417-420; erratum, 741

Polygon matroids for hypergraphs and K-terminal networks, Congressus Numerantium 110 (1995), 153-159

(with J. Benashski ’96, R. Martin and J. Moore) On the beta-invariant for graphs, Congressus Numerantium 109 (1995), 211-221

Crapo’s beta-invariant and K-terminal networks, in Graph Theory, Combinatorics, and Applications: Proceedings of the Seventh Quadrennial International Conference on the Theory and Applications of Graphs, Y. Alavi and K. Schwenk, eds., J. Wiley and Sons, N. Y., 1995, pp. 1169-1177

Generalized activities and K-terminal reliability. II, Discrete Mathematics 135 (1994), 381-385

(with G. Gordon) Polynomials for directed graphs, Congressus Numerantium 94 (1993), 187-201; addendum, ibid. 100 (1994), 5-6

On the star-delta transformation in network reliability, Networks 23 (1993), 151-157

Generalized activities and K-terminal reliability, Discrete Mathematics 96 (1991), 131-149

Higher Fitting invariants, Houston Journal of Mathematics 17 (1991), 55-61

(with G. Gordon) Generalized activities and the Tutte polynomial, Discrete Mathematics 85 (1990), 167-176

A dichromatic polynomial for weighted graphs and link polynomials, Proceedings of the American Mathematical Society 106 (1989), 279-286

Linking numbers and Chen groups, Topology and its Applications 31 (1989), 55-71

Conway’s potential function and its Taylor series, Kobe Journal of Mathematics 5 (1988), 233-264

(with W. S. Massey) On a conjecture of K. Murasugi, Pacific Journal of Mathematics 124 (1986), 193-213

On the Goeritz matrix of a link, Mathematische Zeitschrift 188 (1985), 203-213

Milnor’s invariants and the completions of link modules, Transactions of the American Mathematical Society 284 (1984), 401-424

Some properties of the determinantal ideals of link modules, Mathematics Seminar Notes, Kobe University, 11 (1983), 363-380

(with M. Sakuma) Linking numbers and the groups of links, Mathematics Seminar Notes, Kobe University, 11 (1983), 119-132

Linking numbers and the elementary ideals of links, Transactions of the American Mathematical Society 275 (1983), 309-318

The determinantal ideals of link modules. II, Pacific Journal of Mathematics 109 (1983), 237-245

The determinantal ideals of link modules. I, Pacific Journal of Mathematics 101 (1982), 215-222

A generalization of Torres’ second relation, Transactions of the American Mathematical Society 269 (1982), 593-610

(with W. S. Massey) Links with free groups are trivial, Proceedings of the American Mathematical Society 82 (1981), 155-156