We mathematically analyze the striking visual effect known as planned pooling that arises in knit and crochet patterns when working back and forth with variegated yarn dyed at consistent intervals. Our main result identifies three desirable planned pooling pattern families and provides formulas for choosing row lengths to obtain those patterns.

In this paper we present a method for constructing large, stuffed, machine-knit models of knots and links. Our approach is based on the minimum ropelength of knots, a measure of the minimum amount of rope needed to tie a given knot. We combine known mathematical bounds for minimum ropelength with physical measurements and knit gauge to determine appropriate row counts for creating large-scale stuffed models of particular knots and links on a flatbed knitting machine. The resulting soft sculptures invite tactile, playful exploration of knotted and linked mathematical forms, and are surprisingly huggable.

Graphs are typically represented in published research literature as two-dimensional images, for obvious reasons. With the increased accessibility of 3D rendering software and 3D printing hardware, we can now represent graphs in three dimensions more easily. Years of published work in the field have led to certain ‘standard’ two-dimensional configurations of well-known graphs such as the Petersen graph or K6, but there is no such standard for illustrations of graphs in three-dimensional space. Ideally, a spatial graph configuration should highlight the primary properties and features of the graph, as well as be aesthetically pleasing to view. In this paper, we will suggest and realize standard ideal spatial configurations for a variety of well-known graphs and families of graphs. These configurations can help provide fresh three-dimensional intuition about certain families of graphs, in particular the relationships between graphs in the Y-Delta Petersen family.

In this paper we investigate and enumerate classes of rectangular Celtic knot designs. We introduce criteria for reducing, filtering, and categorizing such designs in order to obtain pleasing, Celtic-looking patterns. We use Hamming graphs to manage large collections of related designs, begin to uncover relationships between Celtic designs, and attempt to identify mathematically and aesthetically significant characteristics of designs.

In many cases, it is possible to use a black box differential equations solver to create printable models of dynamical systems such as chaotic attractors. However, sometimes a black box method is inadequate. We have therefore developed a mixed curvature method to make 3D printing possible in these cases. This mixed curvature methoduses a combination of Matlab and the cost-free software OpenSCAD, designed specifically for 3D mesh export. We mostly restrict our discussion to solutions of differential equations, but we end with an extention to the context of more general dynamical structures for iterated maps and ordinary and partial differential equations.

Mathematicians have established that there are exactly seventeen plane symmetry groups, known as “wallpaper patterns.” Multiple mathematical fiber arts papers describe subsets of those patterns that are suitable for textile design with grid-based media such as knitting and cross-stitch. This paper extends the existing library of these patterns to include orientation-based variations and four-fold rotations. In particular we enumerate all plane lattice symmetries that can be generated from square tiles using lattice translations and combinations of other lattice symmetries. This work has application beyond mathematics as we also provide a useful patterning tool for knitting and fiber arts designers; we introduce Symmetry Generator software created using Python/Processing and OpenSCAD which provides a method for exporting plane lattice symmetry designs as hand knitting patterns or as physical punch cards for use in knitting machines.

Tritangentless knots have a curious and beautiful property: when realized as physical 3D printed models, they roll. Some tritangentless parameterizations roll more easily and freely than others. In this paper we numerically optimize parameters to obtain the most “aesthetically pleasing” rolling knots and then create physical models of these knots using 3D printing, thereby leveraging mathematical tools to obtain an elegant kinetic sculpture.

The game of best choice is a model for sequential decision making with many variations in the literature. Notably, the classical setup assumes that the sequence of candidate rankings are uniformly distributed in time and that there is no expense associated with conducting the candidate interviews. Here, we weight each ranking permutation according to the position of the best candidate in order to model costs incurred from interviews for candidates that are ultimately not hired. We compare our weighted model with the classical model via a limiting process. It turns out that imposing even infinitesimal costs on the interviews results in a probability of success that is about 28%, as opposed to 1/e ≈ 37% in the classical case.

Spiral knots are a generalization of the well-known class of torus knots indexed by strand number and base word repetition. By fixing the strand number and varying the repetition index we obtain integer sequences of spiral knot determinants. In this paper we examine such sequences for spiral knots of up to four strands using a new periodic crossing matrix method. Surprisingly, the resulting sequences vary widely in character and, even more surprisingly, nearly every one of them is a known integer sequence in the Online Encyclopedia of Integer Sequences. We also develop a general form for these sequences in terms of recurrence relations that exhibits a pattern which is potentially generalizable to all spiral knots.

We identify modular-magic Sudoku boards that can serve as representatives for equivalence classes defined from the modular-magic physical symmetries. This will allow us to identify a restricted set of relabeling symmetries that, together with the physical symmetries, forms a minimal complete modular-magic Sudoku symmetry group. We conclude with a simple computation that proves the non-obvious fact that the full Sudoku symmetry group is, in fact, already minimal and complete.

The probabilities, and hence the rankings, of the standard poker hands are well-known. We study what happens to the rankings in a game where a deck is used with a suit missing, or with an extra suit, or extra face cards. In particular, does it ever happen that two or more hands will be equally likely? In this paper we examine this and other questions, and show how probability, some analysis, and even number theory can be applied.

Mancala is a generic name for a family of sowing games that are popular all over the world. There are many two-player mancala games in which a player may move again if their move ends in their own store. In this work, we study a simple solitaire mancala game called Tchoukaillon that facilitates the analysis of “sweep” moves, in which all of the stones on a portion of the board can be collected into the store. We include a self-contained account of prior research on Tchoukaillon, as well as a new description of all winning Tchoukaillon boards with a given length. We also prove an analogue of the Chinese Remainder Theorem for Tchoukaillon boards, and give an algorithm to reconstruct a complete winning Tchoukaillon board from partial information. Finally, we propose a graph-theoretic generalization of Tchoukaillon for further study.

We introduce a new matrix tool for the sowing game Tchoukaillon that enables us to non-iteratively construct an explicit bijection between board vectors and move vectors. This allows us to provide much simpler proofs than currently appear in the literature for two key theorems, as well as a non-iterative method for constructing move vectors. We also explore extensions of our results to Tchoukaillon variants that involve wrapping and chaining.

Calculations of the number of equivalence classes of Sudoku boards has to this point been done only with the aid of a computer, in part because of the unnecessarily large symmetry group used to form the classes. In particular, the relationship between relabeling symmetries and positional symmetries such as row/column swaps is complicated. In this paper we focus first on the smaller Shidoku case and show first by computation and then by using connectivity properties of simple graphs that the usual symmetry group can in fact be reduced to various minimal subgroups that induce the same action. This is the first step in finding a similar reduction in the larger Sudoku case and for other variants of Sudoku.

Spiral knots are a generalization of torus knots we define by a certain periodic closed braid representation. For spiral knots with prime power period, we calculate their genus, bound their crossing number, and bound their m-alternating excess.

This paper uses Gröbner bases to explore the inherent structure of Sudoku puzzles and boards. In particular, we develop three different ways of representing the constraints of Sudoku puzzles with a system of polynomial equations. In one case, we explicitly show how a Gröbner basis can be used to obtain a more meaningful representation of the constraints. Gröbner basis representations can be used to find puzzle solutions or count numbers of boards.

We classify by elementary methods the p-colorability of torus knots, and prove that every p-colorable torus knot has exactly one nontrivial p-coloring class. As a consequence, we note that the two-fold branched cyclic cover of a torus knot complement has cyclic first homology group.

Given a three-dimensional complex algebraic variety with isolated singular point and a sufficiently fine complete resolution of the singularity, we construct an exact sequence of weighted Nash complexes. We use genericity and a theorem from Hironaka to make a careful choice of transverse hyperplane that will define the maps of our exact sequence, and use the properties of the monomial generators of the Nash sheaf to construct a local basis for a certain sheaf of logarithmic 1-forms.

Two Fox m-colorings of a knot or link K are said to be equivalent if they differ only by a permutation of colors. The set of equivalence classes of m-colorings under this relation is the set Cm(K) of Fox m-coloring classes of K. We develop a combinatorical formula for |Cm(K)| for any knot or link K that depends only on the m-nullity of K. As a practical application, we determine the m-nullity, and therefore the value of |Cm(P(p,q,r))|, for any any (p, q, r) pretzel link P(p,q,r).

Every three-dimensional complex algebraic variety with isolated singular point has a resolution factoring through the Nash blowup and the blowup of the maximal ideal over which the second Fitting ideal sheaf is locally principal. In such resolutions one can construct Hsiang-Pati coordinates and thus obtain generators for the Nash sheaf that are the differentials of monomial functions. The results here provide a generalization of the results of Pardon and Stern to the three-dimensional case, as well as a more conceptual view of Pati’s three-dimensional results.

Given a complex algebraic 3-dimensional variety with isolated singular point, we show that there exists a resolution over which the generators of the Nash sheaf can be written locally as the differentials of certain monomial functions. The exponents of these monomials define a sequence of three divisors (the “resolution data”) supported on the exceptional divisor of the resolution. Using these divisors we construct an exact sequence that relates the Nash sheaf to the resolution data. As an application, we use this exact sequence to calculate certain Chern classes of the Nash sheaf, and thus Mather-Chern classes of the variety.