**Student Work**

Each participant in our program produces a paper describing their results. As the program is only eight weeks long, participants don't have much time to revise and polish their papers. When viewing the papers below, please keep in mind that they are not intended to be final versions, but are works in progress. The students have worked hard to produce these results, and have done a great job! The research represented by these papers was made possible by NSF-REU grants DMS-9987955, DMS-0139426, DMS-0453605, DMS-0850959, DMS-1156608, DMS-1461286, DMS-1758020 as well as the generous support of California State University, San Bernardino. Below are some of the papers from each of the topics we have studied: Geometry, and Knot Theory. Although combinatorics is not necessarily one of the topics we still offer as part of this REU, we list the work of our former participants in that field as well.

**Summer 2020 Geometry Projects**

An Examination of Linear Combinations of Skew-Adjoint Build Algebraic Curvature Tensors, by Joey Brundan

k-Plane Scalar Curvature and Structure Groups of Model Spaces, by Preston Kelley

On the geometric realization of irreducible manifolds by decomposable model spaces, by Charlie McMenomy

Assessing the Geometric Realizations of Hermitian Manifolds, by Frank Pryor

Algebraic Requirements for k-Curvature Homogeneous Spaces, by Zoë Smith

**Summer 2020 Knot Theory Projects**

Nested Links, Linking Matrices, and Crushtaceans, by Madeline Brown

Topological Realizations of Fully Augmented Links and Polyhedral Partners, by Lisa Cenek

Crushtaceans and Complements of Fully Augmented and Nested Links, by Matty Zevenbergen

**Summer 2019 Geometry Projects**

General Curvature Homogeneity Theories, by Alex Luna

Canonical Expressions of Algebraic Curvature Tensors, by Katie Ragosta

Linear Independence of Algebraic Curvature Tensors in the Higher-Signature Setting, by Chris Tripp

Decomposable Model Spaces and a Topological Approach to Curvature, by Kevin Tully

A Method to Finding Invariants Using Sectional Curvature, by Montana Williams

**Summer 2019 Knot Theory Projects**

Belted Sum Decompositions in Nested Links, by Maddy Howard

Volume Bounds for Trivalent Planar Graphs, by Anita Koh

Embedded Totally Geodesic Surfaces in Fully Augmented Link Complements, by Sierra Knavel

**Summer 2018 Geometry Projects**

Covariant Derivative Lengths in Curvature Homogeneous Manifolds, by Gabby Konyndyk

An upper bound on \eta(n), by Gabe Lopez

Putting the "k" in Curvature: k-Plane Constant Curvature Conditions, by Maxine Calle

**Summer 2018 Knot Theory Projects**

Systole Length and Preservation Under Belt-Sums of the Borromean Rings, by Amanda Cowell

Factorization of Fully Augmented Links via Belted Sum Decomposition, by Brian Ransom

Volume Bounds of T-links, by Maya Klaib

Geometric Exploration of T-Links using Lorenz Link Correspondence, by Ryan Rhoades

Generalized Cell Decompositions of Nested Lorenz Links, by Wil Coenen

**Summer 2017 Geometry Projects**

Constant Vector Curvature for Skew-Adjoint Linear Transformations , by Mack Beveridge

A complete description of constant vector curvature in the 3-dimensional setting , by Andrew Lavengood-Ryan

Algebraic curvature tensors of Einstein and weakly Einstein model spaces, by Roberta Shapiro

**Summer 2017 Knot Theory Projects**

Belted sum decomposition of fully augmented links, by Porter Morgan, Dean Spyropoulos, Cameron Ziegler (the "Belt Summers")

Geometrically realizing nested torus links, by Amie Bray

Analysis of length spectra of geodesics in the knot complement of the Borromean Rings, by Kayla Neal

**Summer 2016 Geometry Projects**

Constant Vector Curvature in 3-Dimensional Lorentzian Space with Diagonalized Ricci Tensor , by Ana Doktorova

1-Homothety Curvature Homogeneity and 1-Curvature Homogeneity, by Rory Martin-Hagemeyer

A Generalization of Various Theories of Curvature Homogeneity, by Sammy Sbiti

**Summer 2016 Knot Theory Projects**

Cusp Density in Nested Octahedral Links, by Gabrielle Angeloro

Nested and Fully Augmented Links, by Hayley Olson

Iterative Extension on Volume Altering Hyperbolic Tangle Surgeries, by Lucas Everham

Volume Preserving Surgeries on Hyperbolic 3-Manifolds, by Peter Rudzis

Volume Comparison of 3-Braided Links with Dehn Fillings, by Vanessa Vega

**Summer 2015 Geometry Projects**

Curvature Homogeneous Types in Generlized Dunn Manifolds , by Daniel Winney

The Study of the Constant Vector Curvature Curvature Condition for Model Spaces of Three Dimensions in the Lorentzian Setting , by Angela Peng

Structure Groups and Linear Transformations , by Darien Elderfield

Linear Dependence of Canonical Algebraic Curvature Tensors with Associated Chain Complexes, by David Williams

Linear Dependence in Sets of Three Canonical Algebraic Curvature Tensors, by Susan Ye

**Summer 2015 Knot Theory Projects**

Relating Hyperbolic Braids and PSL(2,Z), by Cat Weiss

Fully Augmented Links, by John Harnios

The Bracket Polynomial and Genus of Alternating Links, by Karina Novoa

**Summer 2014 Geometry Projects**

Linearly Dependent Sets of Algebraic Curvature Tensors , by Alberto Acevedo

Linear Dependence of Canonical Algebraic Curvature Tensors of Symmetric and Anti-Symmetric Builds , by Elise McMahon

A Look at Constant Vector Curvature on Three-Dimensional Model Spaces according to the Curvature Tensor , by Albany Thompson

Kernels of Canonical Algebraic Curvature Tensors , by Adam Williams

**Summer 2014 Knot Theory Projects**

On Relating Linking Number and Stick Number , by Jackson Hsu

Three-Variable Bracket Polynomial for Three Braid Knots and Links , by Madeleine Johnson

Recovering Link Data From the B-Graph of an Alternating Diagram , by Maggie Miller

Three-Variable Bracket Polynomial for Algebraic Links , by Tyler Underwood

**Summer 2013 Geometry Projects**

Curvature homogeneity of type (1,3) in pseudo-Riemannian manifolds , by Cullen McDonald

The study of the constant vector curvature condition for model spaces in dimension three with positive definite inner product , by Kelci Mumford

Structure groups of algebraic curvature tensors in dimension three , by Lea Beneish

**Summer 2013 Knot Theory Projects**

Torified Rational Links , by Karly Brint

The three-variable bracket polynomial for reduced, alternating links , by Kelsey Lafferty

Three-variable bracket polynomial for two-bridge knots , by Matthew Overduin

**Summer 2012 Geometry Projects**

Linear Dependence and Hermitian Geometric Realizations of Canonical Algebraic Curvature Tensors , by Dan Diroff

Jordan Normal Forms for a Skew-Adjoint Operator in the Lorentzian Setting , by Lily Saunders

Is Every Invertible Linear Map in the Structure Group of some Algebraic Curvature Tensor? , by Lisa Kaylor

On the Computation and Dimension of Structure Groups of Algebraic Curvature Tensors , by Malik Obeidin

Signature Change and Linear Dependence of Curvature Tensors , by Sarah Schmidt

**Summer 2012 Knot Theory Projects**

Ramsey Numbers and Other Knot Invariants , by Maribeth Johnson

Structure of K Braid Satellites Within a Type K Torus , by Michael Lessley

Proving the absence of certain totally geodesic surfaces , by Warren Shull

**Summer 2011 Geometry Projects**

Linear Independence of Sets of Three Algebraic Curvature Tensors , by Britany Lovell

On the Structure Groups of Decomposable Algebraic Curvature Tensors , by Cole Franks

Linearly Dependent Sets of Algebraic Curvature Tensors with a Nondegenerate Inner Product , by Evan Smothers

An Upper Bound on antisymmetric Algebraic Curvature Tensors , by Joey Thomas

**Summer 2011 Knot Theory Projects**

Volume of Hyperbolic Closed 3-Braids , by Aly Bittner

Linking Number of a Linear Embedding with Two Components , by Julie Wasiuk

Classifying Closed Three- and Four-Braids as Hyperbolic , by Marjorie Kasten

**Summer 2010 Geometry Projects**

Structure Groups of Pseudo-Riemannian Algebraic Curvature Tensors, by Joseph Palmer

Skew-Tsankov Algebraic Curvature Tensors in the Lorentzian setting, by Kaetlin Taylor

Algebraic Curvature Tensors and Antisymmetric Forms, by Blake Treadway

**Summer 2010 Knot Theory Projects**

Second Twist Number of 2-bridge knots, by Jessica Ceniceros

Composing two non-tricolorable knots, by Kelly Harlan

Ropelength of Twists and Alternating Knots, by William Kanegis

Lower Bounds for the Ropelength of Reduced Conformations, by Robert McGuigan

Linear Ropelength upper bounds of twist knots, by Michael Swearingin

**Summer 2009 Geometry Projects**

The linear independence of sets of two and three canonical algebraic curvature tensors, by Alexander Diaz

A Spanning Set for Algebraic Covariant Derivative Curvature Tensors, by Bronson Lim

Constructing Algebraic Curvature Tensors Using Symmetric Bilinear Forms, by John Reynolds

The decomposability of full and weak model spaces, by Danielle Strieby

**Summer 2009 Knot Theory Projects**

Ropelength and Lissajous Diagrams, by Katie Agle

Ropelength, Connect-Sum, and Cabling of Paired Knots, by Jessica Alley

Lower bound for ropelength of m-almost alternating knots, by Charley Mathes

Dowker Notation and Arc Presentation, by Jennifer Winters

**Summer 2007 Knot Theory Projects**

Convexity and Minimum Distance Energy, by Rosanna Speller

Ropelength of Braids and Tangle Decomposition of Knots, by Sarah Ruether

On Ropelength of Alternating Links, by Shahla Sadjadi

**Summer 2007 Combinatorics Projects**

Tree Congestion for Complete n-Partite Graphs, by Allen Cox

Spanning Tree Congestion Critical Graphs, by Daniel Tanner

On Spanning Tree Congestion of Product Graphs, by Rachel Hunter

On 3-cyclic Cutwidth Critical Graphs, by Matthew Straughn

On 3-cyclic Bandwidth Critical Graphs, by Eric Bucher

**Summer 2006 Knot Theory Projects**

The Crossing Probability Knot Energy, by Blake Mitchell

Upper Bound for Ropelength of Pretzel Knots, by Safiya Moran

The Minimum Distance Energy and Midpoint Insertion, by Stephanie Proksch

The Minimum Distance Energy for Polygonal Unknots, by Johanna Tam

**Summer 2006 Combinatorics Projects**

Two-Dimensional Bandwidth, by Chris Duran

Edge Bandwidth, by Taijiro Bernstein

3 and 4-Bandwidth Critical Graphs, by Ann Kilzer

Investigation of 4-Cutwidth Critical Tree Graphs and Complete Cyclic Cutwidth Critical Graphs, by Ruffin Swain

**Summer 2005 Knot Theory Projects**

The Twist Numbers of Graphs and the Tutte Polynomial, by Thomas Albertson

An Upper Bound on Stick Numbers for Links, by Erik Insko

On the Second Twist Number, by Gabriel Murillo

Twist Numbers of Links from the Jones Polynomial, by Matthew Williamson

**Summer 2005 Combinatorics Projects**

The Tree Congestion of Graphs, by Diana Carr

On the Cyclic Cutwidth of Complete Tripartite and n-Partite Graphs, by Heather Allmond

On 3-bandwidth Critical Graphs, by Holly Westerfield

The Linear Cutwidth of Complete n-Partite Graphs, by Chelsea Weitzel

**Summer 2004 Knot Theory Projects**

The Twisted Torus and Knots, by Jenny Buontempo

The Supercrossing Index of Torus Knots, by Ryan Petitfils

Knotted Ribbons and Ribbon Length, by Maria Salcedo

The Stick Number of Tr,4r, by Ivan Ventura

**Summer 2004 Combinatorics Projects**

The Linear Cutwidth of Complete Bipartite and Tripartite Graphs, by Stephanie Bowles

The Linear and Cyclic Wirelength of Complete Bipartite Graphs, by Liz Hartung

On Tree Congestion of Graphs, by Stephen Hruska

Minimum Grid Cutwidth of Complete Bipartite Graphs, by Karl Quick

**Summer 2003 Knot Theory Projects**

Star Move and Construction of a 10-Stick 11-Crossing Knot, by Naomi Davis

The n-iterated Clasp Move and Torus Links, by Stacy Nicole Mills

Iterated Clasp Move and Upper Bounds for 2-bridge Links, by Tim Nawojski

Self-Linking Number as a Geometric Invariant through Geo(8), by Joseph Rhoads

**Summer 2003 Combinatorics Projects**

The Cyclic Cutwidth of Qn, by Jason Erbele

A Proof for the Cyclic Cutwidth of Q6, by Candi Castillo

The Embedding of Complete Bipartite Graphs onto 2´n Grids with Minimum Grid Cutwidth, by Carlos Ibarra

The Cyclic Cutwidth of Complete Bipartite Graphs, by Megan Holben

**Summer 2002 Knot Theory Projects**

On the Lower Bounds of Ramsey Numbers of Knots, by Joe Clark

Knots in a Cubic Lattice, by Marta Kobiela

Stick Numbers of Links with Two Trivial Components, by Alexa Mater

**Summer 2002 Combinatorics Projects**

On Which Graphs Have Linear Cutwidth Equal to Cyclic Cutwidth, by Melissa Banister

The Linear and Cyclic Cutwidth of the Complete Bipartite Graph, by Matt Johnson

Cutwidth of a Complete Graph embedded on an m × n Grid, by Brad Marchand

Proof of the Grid Cutwidth for a Complete Graph, by Alvin Sacdalan

The Cyclic Cutwidth of a P2 × P2 × P n Mesh, by Victor Sciortino

**Summer 2001 Knot Theory Projects**

Clasping Within Polygonal Links, by Adam O'Connor

Minimum Stick Number for Knots and Links, by Bart Podlesny

Upper Bounds for the Stick Number of Three-Braids, by Nerissa Soriano

Upper Bounds for the Stick Numbers of Three Braids, by Diana Wall

**Summer 2001 Combinatorics Projects**

A Proof for the Cyclic Cutwidth of Q5, by Ryan Aschenbrenner

Bandwidths, Cutwidths, and Wirelengths of Complete Graphs of triangular and related numbers embedded on triangular lattice grid host graphs, by LeeAnn Feathers

The cutwidth of the complete graph on 2n vertices when embedded into a 2×n grid, by Annie Wang