## 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

**A little bit about Lorenzo**

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.

Click here to see an advertisement for the circuit theory of 4-regular graphs.

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, VI. Metabelian groups and 2-component links*

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

*Circuit partitions and signed interlacement in 4-regular graphs*

(with D. S. Silver) * Peripheral elements in reduced Alexander modules: an addendum*, Journal of Knot Theory and its Ramifications, to appear

(with D. S. Silver and S. G. Williams) *Core groups*, Journal of Knot Theory and its Ramifications **33 **(2024), article 2450013

*Binary matroids that classify forests*, Algebra and Discrete Mathematics **33 **(2022), 128-150

*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

[Click here for a program discussed in this paper.]

(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

[Click here for the case *n* = 4.]

(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

[Click here for details of the proof of Proposition 7.2.]

*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