Necklace of Triangles 2 Necklace of Triangles 2.jpg
 Necklace of Triangles 2_pre.jpg
This shows an updated version of the necklace of triangles where the triangles may share vertices.
Necklace of Triangles Necklace of Triangles.jpg
 Necklace of Triangles_pre.jpg
This shows an example of a chord diagram I call a "necklace of triangles" as presented in a paper published in the Bridges Conference Proceedings together with its intersection graph.
Tree Bit Tree bit.jpg
 Tree bit_pre.jpg
Many of the diagrams in my research have had a dashed area that represents a section where the intersection graph restricted to this area is a tree. This image shows a zoomed in example of what the dashed area could really look like.
C and Its Mirror Image C and Its Mirror Image.jpg
 C and Its Mirror Image_pre.jpg
This shows the chord diagram C with its mirror image.
The Intersection Graph of C and KT Intersection Graph of C and KT.jpg
 Intersection Graph of C and KT_pre.jpg
This shows the intersection graph of the chord diagrams C and KT as described in other earlier art pieces. The graph is from a paper by B. Mellor. The chord diagrams on the middle two vertices are C and KT themselves, while the other vertices have other chord diagrams that share this intersection graph.
Mirror Image Proof Mirror Image Proof.jpg
 Mirror Image Proof_pre.jpg
This shows a proof that the chord diagram in the top left equals its mirror image, bottom left. Mirror images of purple diagrams are colored yellow.
2 Loops 5 2 loops 5.jpg
2 loops 5_pre.jpg
The chord diagrams are the vertices of the graph and the color of the outside circle matches the chord of the same color inside the diagram that the vertex represents.
2 Loops 4 2 loops 4.jpg
2 loops 4_pre.jpg
The chord diagrams are the vertices of the graph and the color of the outside circle matches the chord of the same color inside the diagram that the vertex represents.
2 Loops 3 2 loops 3.jpg
2 loops 3_pre.jpg
The chord diagrams are the vertices of the graph and the color of the outside circle matches the chord of the same color inside the diagram that the vertex represents.
2 Loops 2 2 loops 2.jpg
2 loops 2_pre.jpg
The chord diagrams are the vertices of the graph and the color of the outside circle matches the chord of the same color inside the diagram that the vertex represents.
2 Loops 1 2 loops 1.jpg
2 loops 1_pre.jpg
The chord diagrams are the vertices of the graph and the color of the outside circle matches the chord of the same color inside the diagram that the vertex represents.
Loop Diagram with Elementary Transformations loop diagram with elementary transformations.jpg
loop diagram with elementary transformations_pre.jpg
The chord diagrams are the vertices of the graph and the color of the outside circle matches the chord of the same color inside the diagram that the vertex represents. There is an edge between vertices if the chords they represent intersect, so this is called the intersection graph of the chord diagram. I can change the diagram in small ways called the elementary transformations that preserve the intersection graph so these are 15 different chord diagrams with this common intersection graph.
Tree Diagram with Elementary Transformations tree diagram of elementary transformations.jpg
tree diagram of elementary transformations_pre.jpg
This is a tree graph drawn as a nature tree. The chord diagrams are the vertices of the graph and the color of the outside circle matches the chord of the same color inside the diagram that the vertex represents. There is an edge between vertices if the chords they represent intersect, so this is called the intersection graph of the chord diagram. I can change the diagram in small ways called the elementary transformations that preserve the intersection graph so these are 15 different chord diagrams with this common intersection graph. Each diagram is one elementary transformation away from those adjacent to it.
Willow willow.jpg
 willow_pre.jpg
This is a tree graph drawn as a willow tree. The vertices are chord diagrams whose intersection graph forms the tree. Vertices have the color on the outside of the circle matching their corresponding chord inside the circle.
Pine pine.jpg
 pine_pre.jpg
This is a tree graph drawn as a pine tree. The vertices are chord diagrams whose intersection graph forms the tree. Vertices have the color on the outside of the circle matching their corresponding chord inside the circle.
Palm palm.jpg
 palm_pre.jpg
This is a tree graph drawn as a palm tree. The vertices are chord diagrams whose intersection graph forms the tree. Vertices have the color on the outside of the circle matching their corresponding chord inside the circle.
Plum plum.jpg
 plum_pre.jpg
This is a tree graph drawn as a plum tree. The vertices are chord diagrams whose intersection graph forms the tree. Vertices have the color on the outside of the circle matching their corresponding chord inside the circle.
C C.jpg
 C_pre.jpg
KT KT.jpg
KT_pre.jpg
9-Wheel Decomposition W9.jpg
W9_pre.jpg
This is the decomposition of the wheel of 9 spokes into chord diagrams, starting at the step with 1 internal vertex.
8-Wheel Decomposition w_8.jpg
w_8_pre.jpg
This is the decomposition of the wheel of 8 spokes into chord diagrams, starting at the step with 1 internal vertex.
7-Wheel Decomposition w_7.jpg
w_7_pre.jpg
This is the decomposition of the wheel of 7 spokes into chord diagrams, starting at the step with 1 internal vertex.
6-Wheel Decomposition w_6.jpg
w6 p1_pre.jpg
This is a decomposition of the wheel with 6 spokes into Chord Diagrams via the STU relations.
$\omega_4^2$ Relation w 4 squared relation.jpg
w 4 squared relation_pre.jpg
This is a visual display of the relation involving $w_4^2$ in the space of 3-graphs found by A. Kaishev.
$\mathcal{A}_5$ and $\mathcal{C}_5$ (1) A5 and C5 (1).jpg
A5 and C5 (1)_pre.jpg
This is an extension I found to the "3-Algebras in Small Degrees" table. This shows the 6/7 of the composite basis elements for degree 5, The 7th composite element I already have in the original table. These formed by multipying all order 4 elements with the order one element and the 2 and 3 primitive elements together, where chord diagram multiplication involves cutting open an arc of each outer cirle and gluing them together. Since open Jacobi diagrams are just closed Jacobi diagrams without the outer circle, I ommitted them in this table extension.
$\mathcal{A}_5$ and $\mathcal{C}_5$ (2) A5 and C5 (2).jpg
A5 and C5 (2)_pre.jpg
These are the final 3 prime elements (out of 10) of my basis for order 5 chord diagrams. I have proved rigourusly that my $\mathcal{A}_5$ basis is indeed a basis. For the$\mathcal{C}_5$ basis I am supposing that since these three diagrams require different prime basis elements in their decomposition they are linearly independent and a basis for$\mathcal{C}_5$.
5-Wheel w5.jpg
w5_pre.jpg
This is a decomposition of the Jacobi Diagram of order 5 that is a wheel with 5 spokes into chord diagrams via the STU and Anti-symmetry relations. This is a natural primitive basis element for$\mathcal{C}_5$, the algebra of closed Jacobi diagrams, and the chord diagrams are basis elements for $\mathcal{A}_5$, the algebra of chord diagrams.
4 Circle Decomposition page 1 4 circle decomposition p1.jpg
4 circle decomposition p1_pre.jpg
4 Circle Decomposition page 2 4 circle decomposition p2.jpg
4 circle decomposition p2_pre.jpg
4 Circle Decomposition page 3 4CD p3.jpg
4CD p3_pre.jpg
This is a decomposition of the Jacobi Diagram of order 5 that has a line of 4 internal circles into chord diagrams via the STU and Anti-symmetry relations. This is a natural primitive basis element for$\mathcal{C}_5$ and this final pages shows the chord diagrams in my discovered basis for $\mathcal{A}_5$.
"$\mathcal{C}_5$ to $\mathcal{A}_5$ final" or "Snowman" C5-A5 final p1.jpg
C5-A5 final p1_pre.jpg
Page 1
"$\mathcal{C}_5$ to $\mathcal{A}_5$ final" or "Snowman" C5-A5 final p2.jpg
C5-A5 final p2_pre.jpg
Page 2
"$\mathcal{C}_5$ to $\mathcal{A}_5$ final" or "Snowman" C5-A5 final p3.jpg
C5-A5 final p3_pre.jpg
This shows the decomposition of this Closed Jacobi Diagram into chord diagrams via the STU and Anti-symmetry relations. Since this diagram requires the basis element of my chosen basis for $\mathcal{A}_5$ that does not show up in the wheel or line of 4 circles, it may serve as a third primitive basis element for$\mathcal{C}_5$.
Order 5 "Swim Goggles" decompostion 5-goggle decomposition p1.jpg
5-goggle decomposition p1_pre.jpg
Page 1
Page 2 5-goggle decomposition p2.jpg
5-goggle decomposition p2_pre.jpg
Page 3 5-goggle decomposition p3.jpg
5-goggle decomposition p3_pre.jpg
Page 4 5-goggle decomposition p4.jpg
5-goggle decomposition p4_pre.jpg
Page 5 5-goggle decomposition p5.jpg
5-goggle decomposition p5_pre.jpg
Page 6 5-goggle decomposition p6.jpg
5-goggle decomposition p6_pre.jpg
Page 7 5GD p7.jpg
5GD p7_pre.jpg
This is the decomposition of the primitive Jacobi diagram of order 5 that kinda looks like swim goggles (or the way I drew it in this draft, a robot head) into chord diagrams via the STU, anti-symmetry and 4-term relations. Upon finding a basis for $\mathcal{A}_5$ this final page shows the decomposition in terms of my choses basis. It is potentially linearly dependent on the diagram with 4 internal circles due to having the same diagrams in the final decomposition.
$t_4$ t4.jpg
t4_pre.jpg
This is the decomposition of the "T" Jacobi diagram of order 4, so call because it looks like a T with several vertical lines. This was an alternative primitive element in the literature and this proves it could have been chosen instead of the diagram with 3 internal circles. In fact, for any order the T can be chosen as an alternative to the diagram with a line of circles as a primitive basis element.
3- Wheel 3 wheel.jpg
3 wheel_pre.jpg
This shows the Jacobi diagram that is a wheel with 3 spokes and its chord diagram decomposition. It happens to be equal to 1/4 the diagram with 2 connected interal circles. This means it would have equally been a good primitive basis element for diagrams of degree 3. This equality does not hold in degree 4, so this coincidence must be due to the fact that we can only have one primitive element in degree 3.
The 3 Algebras in Small Degrees 3 Algebras in Small Degrees final.jpg
3 Algebras in Small Degrees final_pre.jpg
(This is a table showing the bases for the bi-algebras A= The algebra of chord diagrams, C= The algebra of closed Jacobi diagrams, and B= The algebra of open Jacobi diagrams. The 3 algebras are isomorphic so we can see they have the same number of basis elements. Elements above the line for each degree are composed of basis diagrams from smaller degrees and below the line are new diagrams. This table is drawn from the table in Chapter 5 section 9 in the book "Introduction to Vassiliev Knot invariants" by S. Chmutov, S. Duzhin, and J. Mostovoy. A full size version can be seen here.
3 Algebras in Small Degrees blank Colorable Version 3 Algebras in small degrees colorable.jpg
3 Algebras in small degrees colorable_pre.jpg
This is the table for "The 3 Algebras in Small Degrees" before I colored it in. I thought it was a lot of fun to color so I also have this version if anyone wants to print this and color it in for themselves. To print this, choose the poster printing option. Using some sort of photoshop also works to color it, but it doesn't work well in paint.
$\mathcal{C}_4 \to \mathcal{A}_4$ (5) C4-A4 (5).jpg
C4-A4 (5)_pre.jpg
This drawing shows the change of basis via the STU and AS relations going from Closed Jacobi diagrams to chord diagrams.
$\mathcal{C}_4 \to \mathcal{A}_4$ (4) C4-A4 (4).jpg
C4-A4 (4)_pre.jpg
This drawing shows the change of basis via the STU and AS relations going from Closed Jacobi diagrams to chord diagrams.
$\mathcal{C}_4 \to \mathcal{A}_4$ (3) C4-A4 (3).jpg
C4-A4 (3)_pre.jpg
This drawing shows the change of basis via the STU and AS relations going from Closed Jacobi diagrams to chord diagrams.
$\mathcal{C}_4 \to \mathcal{A}_4$ (2) C4-A4 (2).jpg
C4-A4 (2)_pre.jpg
This drawing shows the change of basis via the STU and AS relations going from Closed Jacobi diagrams to chord diagrams.
$\mathcal{C}_4 \to \mathcal{A}_4$ (1) C4-A4 (1).jpg
C4-A4 (1)_pre.jpg
This drawing shows the change of basis via the STU and AS relations going from Closed Jacobi diagrams to chord diagrams.
$\mathcal{C}_3 \to \mathcal{A}_3$ (2) A3-C3(2).jpg
A3-C3(2)_pre.jpg
This drawing shows the change of basis via the STU and AS relations going from Closed Jacobi diagrams to chord diagrams.
$\mathcal{C}_3 \to \mathcal{A}_3$ (1) A3-C3(1).jpg
A3-C3(1)_pre.jpg
This drawing shows the change of basis via the STU and AS relations going from Closed Jacobi diagrams to chord diagrams.
$\mathcal{C}_2 \to \mathcal{A}_2$ A2-C2.jpg
A2-C2_pre.jpg
This drawing shows the change of basis via the STU and AS relations going from Closed Jacobi diagrams to chord diagrams.
Symmeterization Map and its Inverse Symmetrization map and its inverse.jpg
Symmetrization map and its inverse_pre.jpg
The symmeterization map $\Chi$ sends an open Jacobi diagram to the set of closed Jacobi diagrams as sum of all possible attachments of the legs to the Wilson loop (outer circle) divided by the number of possibilities. The map $\tau$ was created as its inverse to prove that the bi-algebra of open Jacobi diagrams is isomorphic to that of closed Jacobi diagrams. This picture shows an open Jacobi diagram going through the symmetrization map and then the inverse to get back to the original diagram.
Intersection Graph Intersection Graph.jpg
Intersection Graph_pre.jpg
The intersection graph of a chord diagram shows which chords intersect eachother. This is the graph that corresponds to the chord diagrams in the nodes. If you flip this diagram along any axis the graph is preserved. I have marked which axis it flips along to get the connecting node by coloring half the node's color and half the connecting nodes color along the circle.
2-T Relations 2-T relations.jpg
2-T relations_pre.jpg
Take a chord diagram and make it into a surface with boundary kinda like it's made of rubber bands and the chords have to go on top of or under one another. We can then trace along the boundary of those bands to find out what our boundary components are. If you look at the diagrams here you can see the boundary components traced in green, purple, pink, and red. The 2-term relations are manipulations of a chord diagram that preserve the number of boundary components. Here I started with the diagram on the upper left and applied 2-T relations to it (or a series of 2-T relations) to get the other diagrams. I have colored the boundary components consistently as they moved around. Modulo these relations the space of chord diagrams as surfaces with boundary has a basis as a "caravan of one and two humped camels". In the bottom right you can see the caravan that all these diagrams are equivalent to modulo the 2-T relations.
$\delta(d^4_5)$ Delta d^4_5.jpg
Delta d^4_5_pre.jpg
This is a drawing of a chord diagram and its coproduct. The coproduct is a map $\delta: A\to A \otimes A$, where A is the space of chord diagrams, that sends a chord diagram to the sum of tensor products of two diagrams where the one on the left has the chords missing from the one on the right. For an order n diagram, there are $2^n$ terms, so for this order 4 diagram, we have the 16 terms shown in the boxes around the edge.
Kontsevich Integral of the Hump Kontsevich Integral of the Hump.jpg
Kontsevich Integral of the Hump_pre.jpg
The Knotsevich Integral is a knot invariant defined as a integral on a knot projected in 3-space viewed as $\mathbb{C}\times \mathbb{R}$. It takes a knot and assigns to it a series of chord diagrams with rational coefficients. This picture depicts the first 9 terms of the (preliminary) Kontsevich Integral for the "Hump" which is the unknot with 2 maxima. You can see the hump depicted in silver and gold in the background. The way the diagrams are colored corresponds to the coefficients, which are +1 -1/24 +1/24 +7/5760+7/2880-17/5760-1/720+1/1920+1/5760. This integral in this explicit form only exists for the unknot (both with and without the bumps).
6-Smile 6-smile.jpg
6-smile_pre.jpg
This is part of a Xmas present series on Intersection Graphs, the original colorings of the pages in my coloring book. The foreground is a graph with chord diagrams on the vertices that have the graph as their intersection graph. (Or in simple terms: the dots are chord diagrams and the way they are connected shows how the lines inside the circles intersect.) This one forms a smiley face.
8-Spiral 8-spiral.jpg
8-spiral_pre.jpg
This is part of a Xmas present series on Intersection Graphs, the original colorings of the pages in my coloring book. The foreground is a graph with chord diagrams on the vertices that have the graph as their intersection graph. (Or in simple terms: the dots are chord diagrams and the way they are connected shows how the lines inside the circles intersect.) This one forms a line which I drew as a spiral.
6-Heart 6-heart.jpg
6-heart_pre.jpg
This is part of a Xmas present series on Intersection Graphs, the original colorings of the pages in my coloring book. The foreground is a graph with chord diagrams on the vertices that have the graph as their intersection graph. (Or in simple terms: the dots are chord diagrams and the way they are connected shows how the lines inside the circles intersect.) This one is a circle I drew as a heart.
5-Peace 5-peace.jpg
5-peace_pre.jpg
This is part of a Xmas present series on Intersection Graphs, the original colorings of the pages in my coloring book. The foreground is a graph with chord diagrams on the vertices that have the graph as their intersection graph. (Or in simple terms: the dots are chord diagrams and the way they are connected shows how the lines inside the circles intersect.) This one forms a peace sign.
5-Complete 5-complete.jpg
5-complete_pre.jpg
This is part of a Xmas present series on Intersection Graphs, the original colorings of the pages in my coloring book. The foreground is a graph with chord diagrams on the vertices that have the graph as their intersection graph. (Or in simple terms: the dots are chord diagrams and the way they are connected shows how the lines inside the circles intersect.) This is a complete graph.
5-Clover 5-clover.jpg
5-clover_pre.jpg
This is part of a Xmas present series on Intersection Graphs, the original colorings of the pages in my coloring book. The foreground is a graph with chord diagrams on the vertices that have the graph as their intersection graph. (Or in simple terms: the dots are chord diagrams and the way they are connected shows how the lines inside the circles intersect.) This one is a line but I drew it as a four leaf clover.
5-Dot in a Square 5-box.jpg
5-box_pre.jpg
This is part of a Xmas present series on Intersection Graphs, the original colorings of the pages in my coloring book. The foreground is a graph with chord diagrams on the vertices that have the graph as their intersection graph. (Or in simple terms: the dots are chord diagrams and the way they are connected shows how the lines inside the circles intersect.)
6-House 6-house.jpg
6-house_pre.jpg
This is part of a Xmas present series on Intersection Graphs, the original colorings of the pages in my coloring book. The foreground is a graph with chord diagrams on the vertices that have the graph as their intersection graph. (Or in simple terms: the dots are chord diagrams and the way they are connected shows how the lines inside the circles intersect.) This one forms a house.
6-Triangle 6-triangle.jpg
6-triangle_pre.jpg
This is part of a Xmas present series on Intersection Graphs, the original colorings of the pages in my coloring book. The foreground is a graph with chord diagrams on the vertices that have the graph as their intersection graph. (Or in simple terms: the dots are chord diagrams and the way they are connected shows how the lines inside the circles intersect.) This one was inspired by the Sierpinski triangle.
7-flower 7-flower.jpg
7-flower_pre.jpg
This is part of a Xmas present series on Intersection Graphs, the original colorings of the pages in my coloring book. The foreground is a graph with chord diagrams on the vertices that have the graph as their intersection graph. (Or in simple terms: the dots are chord diagrams and the way they are connected shows how the lines inside the circles intersect.) This one forms a flower.
7-Jester Hat 7-jester hat.jpg
7-jester hat_pre.jpg
This is part of a Xmas present series on Intersection Graphs, the original colorings of the pages in my coloring book. The foreground is a graph with chord diagrams on the vertices that have the graph as their intersection graph. (Or in simple terms: the dots are chord diagrams and the way they are connected shows how the lines inside the circles intersect.) This one forms a Jester Hat.
10-tree 10-tree.jpg
10-tree_pre.jpg
This is part of a Xmas present series on Intersection Graphs, the original colorings of the pages in my coloring book. The foreground is a graph with chord diagrams on the vertices that have the graph as their intersection graph. (Or in simple terms: the dots are chord diagrams and the way they are connected shows how the lines inside the circles intersect.) This one forms a tree.
10-lizard 10-lizard.jpg
10-lizard_pre.jpg
This is part of a Xmas present series on Intersection Graphs, the original colorings of the pages in my coloring book. The foreground is a graph with chord diagrams on the vertices that have the graph as their intersection graph. (Or in simple terms: the dots are chord diagrams and the way they are connected shows how the lines inside the circles intersect.) This one forms a lizard. It was inspired by my iguana, Lylyth.
8-leaf 8-leaf.jpg
8-leaf_pre.jpg
This is part of a Xmas present series on Intersection Graphs, the original colorings of the pages in my coloring book. The foreground is a graph with chord diagrams on the vertices that have the graph as their intersection graph. (Or in simple terms: the dots are chord diagrams and the way they are connected shows how the lines inside the circles intersect.) This one forms a leaf.
6-diamonds 6-diamonds.jpg
6-diamonds_pre.jpg
This is part of a Xmas present series on Intersection Graphs, the original colorings of the pages in my coloring book. The foreground is a graph with chord diagrams on the vertices that have the graph as their intersection graph. (Or in simple terms: the dots are chord diagrams and the way they are connected shows how the lines inside the circles intersect.)
$10_{42}$ in $\mathcal{A}_5$ 10_42 in A_5.jpg
10_42 in A_5_pre.jpg
This is part of a series I made (for Christmas presents) of 10 crossing knots with 5 double points. The corresponding chord diagram is in the foreground with the knot in the background.
$10_{75}$ in $\mathcal{A}_5$ 10_75 in A_5.jpg
10_75 in A_5_pre.jpg
This is part of a series I made (for Christmas presents) of 10 crossing knots with 5 double points. The corresponding chord diagram is in the foreground with the knot in the background.
$10_{92}$ in $\mathcal{A}_5$ 10_92 in A_5.jpg
10_92 in A_5_pre.jpg
This is part of a series I made (for Christmas presents) of 10 crossing knots with 5 double points. The corresponding chord diagram is in the foreground with the knot in the background.
Knot in $\mathcal{A}_5$ Knot in A_5.jpg
Knot in A_5_pre.jpg
In the background is a singular knot with 5 double points and in the foreground is the corresponding chord diagram.
Singularization of $10_{10}$ Singularization of 10_10.jpg
Singularization of 10_10_pre.jpg
This is a depiction of the $10_{10}$ knot with 3 double points added, as represented by spirals on some of the crossings. This was part of a series made as Christmas presents in 2013.
Singularization of $10_{28}$ Singularization of 10_28.jpg
Singularization of 10_28_pre.jpg
This is a depiction of the $10_{28}$ knot with 3 double points added, as represented by spirals on some of the crossings. This was part of a series made as Christmas presents in 2013.
Singularization of $10_{122}$ Singularization of 10_122.jpg
Singularization of 10_122_pre.jpg
This is a depiction of the $10_{122}$ knot with 2 double points added, as represented by spirals on some of the crossings. This was part of a series made as Christmas presents in 2013.
Singularization of $10_{132}$ Singularization of 10_132.jpg
Singularization of 10_132_pre.jpg
This is a depiction of the $10_{132}$ knot with 2 double points added, as represented by spirals on some of the crossings. This was part of a series made as Christmas presents in 2013.
$\Sigma_{12}$ Butterfly Sigma 12 Butterfly.jpg
Sigma 12 Butterfly_pre.jpg
A singular knot with 12 double points in the shape of a butterfly.
The Embedding of the Circle in $\mathbb{R}^3$ as the $7_4$ Knot The Embedding of S^1 in R^3 as 7_4.jpg
The Embedding of S^1 in R^3 as 7_4_pre.jpg
An embedding is a one to one continuous map of a smaller manifold into a bigger one. Here, we can embed any circle into three space as any knot. The border "circle" and the knot are colored to show the one to one match.
Dyadic Ideal Triangles on the Poincare Disk Dyadic Ideal Triangles on the Poincare Disk.jpg
Dyadic Ideal Triangles on the Poincare Disk_pre.jpg
The circle is the Poincare Disk model of hyperbolic space. Geodesics (strait lines) are diameters of the circle and circles intersecting the boundary circle at right angles. An Ideal Triangle is one with 0-degree angles as you can obtain on this model when vertices are touching the boundary circle. While the triangles appear smaller and smaller, they are congruent in hyperbolic space.
Orientable Identifications of Hexagons Orientable identifications of hexagons.jpg
Orientable identifications of hexagons_pre.jpg
If you identify the sides of a hexagon you can glue it together to make either a sphere or a torus. This shows the various idenifications of the hexagon overlayed with eachother. Identified sides are the same color with a bar in the middle connecting them. Vertices that match are the same color. If there are two vertices the identification gives a torus, four vertices gives a sphere.
$S_4$ S_4.jpg
S_4_pre.jpg
This piece depicts the symmetric group on 4 elements. Each element is the product of the piror 2 going around in a spiral starting at the top left. (Although, I may have gotten the order backwards so that it really starts in the 2nd row 3rd column.) The black or white borders represent the sign of the permutation, black for odd, white for even. The identity is the upper left. The background colors in each column are the four rotations with the last column being the cycle (1 2).
Real Analysis According to Dr. Bob Burton Real Analysis According to Bob Burton.jpg
Real Analysis According to Bob Burton_pre.jpg
When I arrived at graduate school I had a conversation with Dr. Burton about how I liked Real Analysis in undergrad. He replied that Real Analysis was essentially you take an open set in $\mathbb{R}^n$ and you take the unit cubes. If a cube fits in you put it in, and if it's outside throw it out, then go down a half size. If one of these cubes fits in put it in and if it fits out throw it out and keep repeating this process. So, in this picture I took my open set, a clover sketched with plates, and started with 6-inch squares. If they fit in I colored them, if I threw them out I made them grayscale.
Hill ♥ Will Hill Heart Will.jpg
Hill Heart Will_pre.jpg
This piece was made for my sister and her new husband as a wedding present. She is a chemistry teacher. The twist knot has clasped ends made to look like Erlenmeyer flasks pouring out their respective favorite colors. In the lower right hand corner is the chemical symbol for Oxytocin, the love chemical.
Amy and Evan Amy and Evan.jpg
Amy and Evan_pre.jpg
This piece was made as a wedding present for my friends Amy and Evan. It is a 3-colorable twist knot linked as a Hopf-link with a circle. The colors are the favorite colors of the couple.
Borromean linked knots Borromean linked knots.jpg
Borromean linked knots_pre.jpg
This is the $9_1$ knot, a 12-crossing twist knot, and a circle linked like the Borromean rings. The knots are 3-colored.
Inspired by the Proof of the Heine-Borel Theorem Inspired by the proof of the Heine-Borel Theorem.jpg
Inspired by the proof of the Heine-Borel Theorem_pre.jpg
The Heine-Borel theorem states that any closed, bounded subset of $\mathbb{R}^n$ is compact. The proof shows that such a subset can be covered by finitely many ball of smaller and smaller radius. Here we have a piece of paper which is a closed and bounded subset of the plane. For artistic reasons I covered it with tangental circles instead of overlaping ones. Each circle covering is made of smaller and smaller radius and is overalyed without regard to any other cover. This piece is on display in the math hearth at Willamette University.
Fibonacci Knots Fibonacci Knots_2.jpg
Fibonacci Knots_2_pre.jpg
This is a family of (2,n)-torus links with the Fibonacci sequence number of crossings (1-crossing links being the unknot since they are trivial). In the background is a golden ratio spiral repeatedly outlined in a rainbow pattern. This piece is on display in the Math Hearth at Willamette University.
Fundamental 3-colorings of a (3,6,-3) pretzel knot Fundamental 3-colorings of a (3,6,-3) pretzel knot.jpg
Fundamental 3-colorings of a (3,6,-3) pretzel knot_pre.jpg
This picture displays all 4 non-trivial 3 colorings of this knot. It was made as part of a final project for my undergrad Topology class taught by Inga Johnson. It is on display in the math hearth at Willamette University.
$D_6$ Tiling D_6 Tiling.jpg
D_6 Tiling_pre.jpg
This is a tiling of the dihedral group of symmetries of a hexagon. Along the diagonal from upper left to bottom right each element is the product of the prior 2 elements. This is also the case down the 2nd and 4th column. The identity element is the uppermost full hexagon in the 3rd column.
Rainbow Radii Rainbow Radii.jpg
Rainbow Radii_pre.jpg
This piece was inspired by talking about sequences converging by being within balls of decreasing radii. I layed out crayons in a rainbow order and started drawing lines of increasing length, spiraling around until the lines filled the page. I have since made many pieces in a similar fashion, but this is the original.
Rainbow Spiral Mural Rainbow Spiral Mural.jpg
Rainbow Spiral Mural_pre.jpg
This was a full wall mural painted in the apartment where I lived my senior year of college. I started with a short red line and slowly added yellow in to draw subsequent lines until I cirled around the rainbow and drew enough lines to fill the wall. This is based on "Rainbow Radii".
Tribute to Dorf 401 Tribute to Dorf 401.jpg
Tribute to Dorf 401_pre.jpg
This is a painting on canvass made in the same fashion as "Rainbow Spiral Mural". It starts with a short red lines and paint was mixed in to go around the rainbow until the lines filled the canvass.
(Red)$\times$(Yellow)$\times$(Blue) (Red)X(Yellow)X(Blue).jpg
(Red)X(Yellow)X(Blue)_pre.jpg
This is a radii spiral sequence where I started at 3 points with Red Yellow and Blue and then spiraled them around.
(Red,...)$\times$(Yellow,...)$\times$(Blue,...) (Red, ...)X(Yellow, ...)X(Blue, ...).jpg
(Red, ...)X(Yellow, ...)X(Blue, ...)_pre.jpg
In this radii spiral drawing I started a rainbow fade at the 3 primary colors and went around from there. The name is in reference to the fact that there are 3 sequences of radii spirals starting at each of those colors.
Dual Radii Spiral Dual Radii Spiral.jpg
Dual Radii Spiral_pre.jpg
This is a radii spiral picture which starts as opposing complementary colors.
April April.jpg
April_pre.jpg
This is one of my radii spiral drawings. It is a rainbow fade starting at blue.
May May.jpg
May_pre.jpg
Another radii spiral drawing with a rainbow fade.
$\widetilde{\text{green}}$ spiral thingy.jpg
spiral thingy_pre.jpg
Another radii spiral drawing. This one starts at 2 points, oil pastel and crayon, at spring green then goes around the rainbow.
$7_4$ 7_4.jpg
7_4_pre.jpg
This is a drawing of the $7_4$ knot (numbered by the Rolfsen Knot Table). The knot is oriented with a rainbow fade done in oil pastel and the background is chalk pastel of similar colors.
$8_{18}$ 8_18.jpg
8_18_pre.jpg
This is a drawing of the $8_{18}$ knot oriented with 4 rainbow fades. Note: this is a poor quality photo of this, but it is the only picture I have. This piece is much brighter in real life.
$9_{35}$ 9_35.jpg
9_35_pre.jpg
This is a picture of the $9_{35}$ knot with one of its 3-colorings.
$9_{40}$ 9_40.jpg
9_40_pre.jpg
This is a depiction of the $9_{40}$ knot oriented by two rainbow fades.
$9_{47}$ 9_47.jpg
9_47_pre.jpg
This is a painting of a symmetric view of the $9_{47}$ knot. Each arc is a different color. Knot is painted with crayon background.
$10_{116}$ 10_116.jpg
10_116_pre.jpg
Depiction of the $10_{116}$ knot using oil pastel.
$10_{155}$ 10_155.jpg
10_155_pre.jpg
Depiction of the $10_{155}$ knot in acrylic paint with each arc a different color.
13-crossing heart knotwith indices identified 13-crossing heart knot with indices identified.jpg
13-crossing heart knot with indices identified_pre.jpg
This knot projection divides the plane into regions. The index of one of these regions is the net number of times the projection winds counterclockwise around the region. The knot is oriented with a rainbow fade and the same colors used in the reverse way in a given region demonstrate the negative of the index.
Borromean Rings Boromean Rings.jpg
Boromean Rings_pre.jpg
This is a picture of the Borromean rings. Each component painted a primary color. Crayon is used for the background.
Closed 3-Braid knot family Closed 3-Braid knot family.jpg
Closed 3-Braid knot family_pre.jpg
This is a family of knots I noticed in the table. They aren't quite torus knots but their projections are similar to (3,n)-torus knots.
Closed Braids Closed Braids.jpg
Closed Braids_pre.jpg
This is a family of (2,n)-torus links (odd crossings are knots, but even crossings are 2-component links).
Fundamental 3-colorings of the $9_{35}$ Knot Fundamental 3-colorings of 9_35 knot.jpg
Fundamental 3-colorings of 9_35 knot_pre.jpg
This picture shows all 4 nontrivial 3-colorings of this knot.
Rainbow Twists Rainbow Twists.jpg
Rainbow Twists_pre.jpg
Chain link Chain link.jpg
Chain link_pre.jpg
Reflections of the Happy Parallelogram Reflections of the Happy Parallelogram.jpg
Reflections of the Happy Parallelogram_pre.jpg
This piece was created by reflecting and translating a parallelogram I made out of paper. The spiral inverts and which colors were oil pastel and which ones were crayon switch between reflections. Translations are colored differently but have the same spiral direction. This was created as a gift to a student I tutored in Geometry, based on our reflection/translation lesson.
Triangle Triangle.jpg
Triangle_pre.jpg
Square Square.jpg
Square_pre.jpg
Hexagon Hexagon.jpg
Hexagon_pre.jpg
Rainbow Waves Rainbow Waves.jpg
Rainbow Waves_pre.jpg
5 5.jpg
5_pre.jpg
Here we have 5 spirals going one direction and 5 going the other direction. The space created by the intersections is colored in alternating red and blue dots.
6 6.jpg
6_pre.jpg
Similar to "5", 6 spirals going one direction were drawn and 6 going the other direction. The space created by their intersections is colored with orange and brown dots. (I created this as a present for a friend and those were the colors she liked.)
Fibonacci Spirals with Dots Fibonacci spirals with dots.jpg
Fibonacci spirals with dots_pre.jpg
This was an attempt at drawing several opposing spirals to mimic the pine-cone pattern. I filled it in with rainbow dots.
Shattered Daisies Shattered Daisies.jpg
Shattered Daisies_pre.jpg
Two daisies of different colors woven into each other.
Copy of Two Cypresses by Vincent Van Gogh Copy of Two Cypresses by Vincent Van Gogh.jpg
Copy of Two Cypresses by Vincent Van Gogh_pre.jpg
This was my final art project for an art class I took while studying abroad at the Universidad de San Francisco de Quito in Ecuador. It is done completely in Crayola crayon.
Tribute to the Fibonacci Sequence Tribute to the Fibonacci Sequence.jpg
Tribute to the Fibonacci Sequence_pre.jpg
I started in the center with a circle of one color, then one color, then 2 colors... then 55 colors. 89 colors would have been too much so I started again with one blue circle around this center collection. Below that 2 pink circles, 3 green circles and so on.
Tribute to the Rainbow in Banos Tribute to the Rainbow in Banos.jpg
Tribute to the Rainbow in Banos_pre.jpg
This picture shows a mountain (green dots) with a rainbow and grey dots for the sky. It was inspired by a rainbow I saw while in Banos, Ecuador.
Dots Dots Part 1.jpg
Dots Part 1_pre.jpg
Galapagos Islands Galapagos islands.jpg
Galapagos islands_pre.jpg
This picture depicts the different kinds of sand on the beaches and different colors of water near the shores of the various Galapagos Islands. In the center is a lava lizard.
Happy Pattern Happy Pattern.jpg
Happy Pattern_pre.jpg
Infinite Hall Infinite Hall.jpg
Infinite Hall_pre.jpg
A prespective drawing experiment.
Invertible Cube Pattern Invertible Cube Pattern.jpg
Invertible Cube Pattern_pre.jpg
Mindo Mindo.jpg
Mindo_pre.jpg
A tribute to the cloud forest town in Ecuador.
Spiral Spiral.jpg
Spiral_pre.jpg