A Survey of Knot Mosaics Research

1. Introduction

Knot theory is a fascinating branch of topology that studies mathematical knots, which are embeddings of circles in three-dimensional Euclidean space. While knot theory has a rich history dating back to the 19th century, the specific concept of knot mosaics is relatively recent, introduced by Samuel J. Lomonaco and Louis H. Kauffman in 2008. This innovative approach to representing knots has opened up new avenues for research and has connections to quantum computation and quantum physics.

This survey aims to provide a comprehensive introduction to the field of knot mosaics for an undergraduate student preparing to work with Professor Lew Ludwig during a summer research project. The survey covers the foundational concepts, historical development, key research results, major contributors to the field, and open questions that might guide future research.

Knot mosaics offer a discrete, combinatorial approach to knot theory that is particularly accessible to undergraduate researchers. By representing knots on a square grid using a finite set of tiles, knot mosaics provide a concrete and visual way to study knot properties. This approach has led to new invariants, such as the mosaic number, and new questions about the relationship between these invariants and classical knot invariants like the crossing number.

The field of knot mosaics has seen significant development since its introduction in 2008, with contributions from researchers like Kuriya and Shehab, who proved the equivalence of tame knot theory and mosaic knot theory, and H.J. Lee, K. Hong, H. Lee, and S. Oh, who established important relationships between mosaic number and crossing number. Professor Lew Ludwig has made substantial contributions to the field, particularly in determining the mosaic number for various classes of knots and in identifying an infinite family of knots whose mosaic number is realized only in non-minimal crossing representations.

As an undergraduate student preparing to work with Professor Ludwig, this survey will provide you with the necessary background to understand the current state of knot mosaics research and to identify potential areas for your own contributions. The field offers many opportunities for undergraduate research, from tabulating mosaic numbers for specific knots to exploring theoretical questions about the relationship between mosaic number and other knot invariants.

2. Foundational Concepts

2.1 Definition of Knot Mosaics

A knot mosaic is a representation of a knot or link on a square grid using a specific set of tiles. Formally, as defined by Lomonaco and Kauffman in 2008, an n-mosaic is an n × n matrix M = (Mij) of mosaic tiles, where each Mij is an element of the set of 11 mosaic tiles.

The concept was introduced as part of a larger framework for quantum knot systems, where a quantum knot is a quantum superposition of these classical knot mosaics. However, the classical knot mosaic system itself has proven to be a rich area for mathematical research.

A key property of knot mosaics is that they provide a discrete, combinatorial approach to knot theory. This makes certain questions about knots more tractable and opens up new avenues for research, particularly in relation to computational approaches to knot theory.

2.2 The 11 Mosaic Tiles

The knot mosaic system uses 11 basic tiles, which are the building blocks for constructing any knot mosaic. These tiles are:

T₀: The empty tile (containing no curve segments)

T₁-T₄: Four tiles with a single curve segment connecting two adjacent sides

T₅-T₈: Four tiles with two curve segments, each connecting two adjacent sides

T₉-T₁₀: Two tiles representing crossings (overcrossing and undercrossing)

These 11 tiles are sufficient to represent any tame knot as a knot mosaic, as proven by Kuriya and Shehab in their work on the Lomonaco-Kauffman Conjecture.

The connection points on these tiles are located at the midpoints of the tile edges. When tiles are placed adjacent to each other on the mosaic grid, these connection points must align to form continuous curves representing the knot.

2.3 Mosaic Moves and Equivalence

Just as in traditional knot theory, where Reidemeister moves establish equivalence between different diagrams of the same knot, the knot mosaic system has its own set of moves that establish equivalence between different mosaic representations of the same knot.

Lomonaco and Kauffman defined several types of mosaic moves:

1. Planar isotopy moves: These moves preserve the topological structure of the knot while changing its geometric representation within the mosaic.
2. Mosaic Reidemeister moves: These are the mosaic equivalents of the traditional Reidemeister moves in knot theory. They include:
   • Type I moves: Adding or removing a twist
   • Type II moves: Adding or removing two crossings
   • Type III moves: Moving a strand across a crossing

The ambient isotopy in knot mosaic theory is generated by these mosaic moves. Two knot mosaics are considered equivalent if one can be transformed into the other through a sequence of these moves.

An important concept in knot mosaic theory is the mosaic number of a knot, denoted m(K), which is the smallest integer n such that the knot K can be represented as an n × n knot mosaic. This invariant has been the focus of much research in the field, particularly in relation to other knot invariants such as the crossing number.

The mosaic system also introduces the concept of suitably connected, which means that each tile in the mosaic must have all of its connection points connected to adjacent tiles, except possibly at the boundary of the mosaic. This ensures that the mosaic represents a proper knot or link without any loose ends.

3. Historical Development

3.1 Origins: Lomonaco and Kauffman (2008)

The concept of knot mosaics was first introduced by Samuel J. Lomonaco and Louis H. Kauffman in their 2008 paper "Quantum Knots and Mosaics," published in the journal Quantum Information Processing. This groundbreaking paper established the foundation for what would become a vibrant area of research in knot theory.

Lomonaco and Kauffman's original motivation was to create a "quantum embodiment" of a closed knotted physical piece of rope. They sought to define a quantum knot system whose states, called quantum knots, would represent the state of a knotted closed piece of rope. This quantum system would allow for non-classical behavior such as quantum superposition and quantum entanglement, raising interesting questions about the relationship between topological and quantum entanglement.

The paper introduced several key concepts:

1. The 11 mosaic tiles that form the building blocks of knot mosaics
2. The definition of an n×n knot mosaic as an arrangement of these tiles
3. The ambient group of unitary transformations representing ways of moving the knot
4. The concept of quantum knots as states of a quantum knot system

While the quantum aspects of their work opened up fascinating theoretical questions, the classical knot mosaic system they defined as a foundation has itself become a rich area for mathematical research.

3.2 Equivalence to Tame Knot Theory

One of the most significant early developments in knot mosaic theory came from the work of T. Kuriya (2008) and O. Shehab (2012), who together proved what is known as the Lomonaco-Kauffman Conjecture. This conjecture stated that tame knot theory and knot mosaic theory are equivalent.

Their proof, published in 2014 in a paper titled "The Lomonaco-Kauffman Conjecture," established that knot mosaic type is a complete invariant of tame knots. This means that any tame knot can be represented as a knot mosaic, and two knot mosaics represent the same knot if and only if they are of the same knot mosaic type (i.e., one can be transformed into the other through a sequence of mosaic moves).

This result was crucial for the development of knot mosaic theory, as it established that the mosaic system is not just a curiosity but a legitimate alternative framework for studying knots. It confirmed that any result derived in the context of knot mosaics has a direct interpretation in traditional knot theory.

Following this foundational work, the field of knot mosaics began to expand in several directions:

1. Tabulation efforts: Researchers began working to determine the mosaic number for various classes of knots, similar to the tabulation efforts in traditional knot theory.
2. Relationship to other invariants: Studies explored the relationship between the mosaic number and other knot invariants, particularly the crossing number.
3. Efficiency questions: Researchers investigated questions about the minimum size of mosaic needed to represent a given knot and the minimum number of non-blank tiles required.
4. Extensions and generalizations: The basic mosaic system was extended and generalized in various ways, such as the introduction of corner connection tiles by Heap et al. in 2023.

These developments have established knot mosaics as a vibrant and growing area of research within knot theory, with connections to both pure mathematics and potential applications in quantum physics and quantum computation.

4. Key Developments and Results

4.1 Relationship Between Mosaic Number and Crossing Number

One of the most significant areas of research in knot mosaics has been exploring the relationship between the mosaic number m(K) and the crossing number c(K) of a knot K. This question was originally posed by Lomonaco and Kauffman in their 2008 paper.

H.J. Lee, K. Hong, H. Lee, and S. Oh made substantial progress on this question in their 2014 paper "Mosaic Number of Knots." They established important bounds:

• For nontrivial knots (except the Hopf link and the knot 6₃): m(K) ≤ c(K)+1
• For prime and non-alternating knots: m(K) ≤ c(K)-1

These bounds provide a direct connection between the mosaic number, a relatively new invariant specific to knot mosaics, and the crossing number, one of the most fundamental invariants in classical knot theory.

Additional relationships were established by other researchers:

• Bae-Park (2000) showed that for knots or non-split links: a(K) ≤ c(K)+2, where a(K) is the arc index
• Jin-Park (2010) demonstrated that for non-alternating prime knots or links: a(K) ≤ c(K)

These results, combined with the relationship between arc index and mosaic number, have helped to establish a more complete understanding of how these different knot invariants relate to each other.

4.2 Tabulation of Knot Mosaics

A significant effort in knot mosaics research has been the tabulation of mosaic numbers for various classes of knots, similar to the tabulation efforts in traditional knot theory.

Lee, Ludwig, Paat, and Peiffer made a major contribution in this area with their 2016 paper "Knot Mosaic Tabulation," published in Involve in 2018. They determined the mosaic number for all eight-crossing or fewer prime knots. This work was particularly notable for being written at an introductory level to encourage undergraduate researchers to explore the topic.

Building on this work, Heap, Baldwin, Canning, and Vinal extended the tabulation to prime knots with crossing number 10 or less in their 2023 paper "Tabulating Knot Mosaics: Crossing Number 10 or Less." They implemented an algorithmic programming approach to find both the mosaic number and tile number for these knots.

These tabulation efforts have not only provided valuable data for further research but have also made the field more accessible to undergraduate researchers by providing concrete examples and results.

4.3 Space-Efficient Knot Mosaics

Another important development in knot mosaics research has been the concept of space-efficient knot mosaics, introduced by Heap and Knowles in their 2018 paper "Tile Number and Space-Efficient Knot Mosaics."

The tile number of a knot mosaic is the minimum number of non-blank tiles needed to represent a knot on a mosaic of a given size. A space-efficient knot mosaic is one that uses the minimum possible number of non-blank tiles for a given knot.

Heap and Knowles explored the relationship between tile number and mosaic number, establishing important results about the efficiency with which knots can be represented on mosaics.

This line of research was extended in subsequent papers:

• Heap and Knowles (2019) focused on space-efficient representations of prime knots with mosaic number 6
• Heap and LaCourt (2020) extended the study to prime knots with mosaic number 7

These studies of space efficiency have added another dimension to knot mosaic research, considering not just the size of the mosaic but also the number of tiles used within that mosaic.

4.4 Non-Reduced Projections

An intriguing question in knot mosaic theory is whether the mosaic number of a knot is always realized in a minimal crossing representation of the knot. Ludwig, Evans, and Paat addressed this question in their 2013 paper "An Infinite Family of Knots Whose Mosaic Number is Realized in Non-Reduced Projections."

They answered a question posed by Colin Adams by constructing an infinite family of knots whose mosaic number can be realized only when the crossing number is not minimal. This was the first known infinite family of non-minimal knots whose mosaic numbers are known.

This result is significant because it shows that the relationship between mosaic number and crossing number is more complex than might initially be expected. It demonstrates that sometimes the most efficient way to represent a knot as a mosaic requires using a non-minimal crossing representation of the knot.

4.5 Corner Connection Tiles

A recent innovation in knot mosaics is the introduction of corner connection tiles by Heap et al. in their 2023 paper "Knot Mosaics with Corner Connection Tiles."

Traditional knot mosaic tiles have connection points at the midpoints of the tile edges. Heap et al. introduced a new set of tiles where the connection points are located at the corners of the tiles instead. They demonstrated that these corner connection tiles can create more efficient knot mosaics for knots with small crossing numbers.

Specifically, they showed that all knots with crossing number 8 or less can be represented on mosaics that use fewer non-blank tiles with corner connection tiles than is possible with traditional tiles.

This innovation opens up new possibilities for representing knots more efficiently and raises interesting questions about how different tile sets might affect the mosaic number and tile number of knots.

5. Key Researchers and Their Contributions

5.1 Founding Researchers

Samuel J. Lomonaco (University of Maryland Baltimore County)

Samuel J. Lomonaco is one of the co-creators of the knot mosaics concept. His background in quantum computation and knot theory led to the development of quantum knot systems, with knot mosaics serving as the foundational representation. Lomonaco's work has focused on:

• The quantum embodiment of knots through mosaic representations
• Exploring the relationship between topological and quantum entanglement
• Developing the formal system of knot mosaics as a rewriting system

Lomonaco's extensive work in quantum computation provided the motivation for creating a mathematical framework that could potentially model quantum phenomena using knot theory.

Louis H. Kauffman (University of Illinois at Chicago)

Louis H. Kauffman, the other co-creator of knot mosaics, is a renowned knot theorist best known for his work on knot polynomials, particularly the Kauffman polynomial and bracket polynomial. His contributions to knot mosaics include:

• Developing the 11 mosaic tiles that form the building blocks of knot mosaics
• Defining the ambient group of transformations for knot mosaics
• Exploring quantum observables that are invariants of quantum knot type

Kauffman's expertise in knot invariants and quantum topology has been instrumental in establishing the theoretical foundations of knot mosaics and their potential applications in quantum physics.

5.2 Major Contributors

T. Kuriya and O. Shehab

Kuriya and Shehab made a fundamental contribution to knot mosaic theory by proving the Lomonaco-Kauffman Conjecture, which established that tame knot theory and mosaic knot theory are equivalent. Their 2014 paper "The Lomonaco-Kauffman Conjecture" demonstrated that:

• Any tame knot can be represented as a knot mosaic
• Knot mosaic type is a complete invariant of tame knots

This result was crucial for validating the knot mosaic approach as a legitimate alternative framework for studying knots.

H.J. Lee, K. Hong, H. Lee, and S. Oh

This group of researchers made significant contributions to understanding the relationship between mosaic number and crossing number. Their 2014 paper "Mosaic Number of Knots" established important bounds:

• For nontrivial knots (except Hopf link and 6₃): m(K) ≤ c(K)+1
• For prime and non-alternating knots: m(K) ≤ c(K)-1

These bounds provide a direct connection between the mosaic number and the crossing number, one of the most fundamental invariants in classical knot theory.

Aaron Heap (SUNY Geneseo)

Aaron Heap has been a prolific contributor to knot mosaics research, with a focus on space efficiency and tabulation. His contributions include:

• Creating the Knot Mosaic Space online repository
• Introducing the concept of space-efficient knot mosaics
• Developing algorithmic approaches to knot mosaic tabulation
• Introducing corner connection tiles as an alternative to traditional mosaic tiles

Heap's work has significantly expanded the practical aspects of knot mosaics, making the field more accessible through online tools and comprehensive tabulations.

5.3 Professor Lew Ludwig's Work

Professor Lew Ludwig of Denison University has made substantial contributions to knot mosaics research, particularly in the areas of mosaic number determination and non-reduced projections.

Key Publications

Ludwig's 2013 paper with Evans and Paat, "An Infinite Family of Knots Whose Mosaic Number is Realized in Non-Reduced Projections," answered a question posed by Colin Adams by constructing an infinite family of knots whose mosaic number can be realized only when the crossing number is not minimal. This was the first known infinite family of non-minimal knots whose mosaic numbers are known.

His 2016 paper with Lee, Paat, and Peiffer, "Knot Mosaic Tabulation," determined the mosaic number for all eight-crossing or fewer prime knots. This work was written at an introductory level to encourage undergraduate researchers to explore the topic.

Research Focus

Ludwig's research in knot mosaics has focused on several key areas:

1. Mosaic Number Determination: Working on determining the mosaic number for various classes of knots and contributing to the tabulation of mosaic numbers for prime knots.
2. Non-Reduced Projections: Investigating cases where the mosaic number of a knot is realized only in non-minimal crossing representations and proving the existence of an infinite family of such knots.
3. Undergraduate Research Mentorship: Engaging undergraduate students in knot mosaics research, co-authoring papers with undergraduate researchers, and creating accessible materials to introduce students to the field.

Educational Contributions

Ludwig has made significant contributions to making knot mosaics accessible as an educational tool through presentations, lectures, and contributions to the Encyclopedia of Knot Theory (2021).

His ongoing research continues to explore open questions in knot mosaics, including the relationship between mosaic number and crossing number, the mosaic number of infinite families of knots, and the properties of knots whose mosaic representations require non-minimal crossing numbers.

6. Open Questions and Future Directions

6.1 Foundational Questions

Relationship Between Mosaic Number and Crossing Number

While significant progress has been made in understanding the relationship between mosaic number m(K) and crossing number c(K), several questions remain:

• Are the established bounds (m(K) ≤ c(K)+1 for nontrivial knots and m(K) ≤ c(K)-1 for prime and non-alternating knots) tight?
• Are there specific families of knots for which more precise relationships can be established?
• Can we characterize exactly when m(K) = c(K), m(K) = c(K)+1, or m(K) = c(K)-1?

Quantum Knot System Applications

Lomonaco and Kauffman's original motivation for developing knot mosaics was to create a quantum knot system. Several questions about the quantum aspects remain open:

• Can quantum knots be used to simulate and predict the behavior of quantum vortices in liquid helium II and Bose-Einstein condensates?
• Might quantum knots provide a mathematical model for gaining insight into charge quantification in the fractional quantum Hall effect?
• How can the formalism of quantum knots be implemented in actual quantum systems?

6.2 Questions About Infinite Families of Knots

Mosaic Number Realization

Ludwig, Evans, and Paat identified an infinite family of knots whose mosaic number is realized only in non-minimal crossing representations. This raises several questions:

• Are there other infinite families with this property?
• Is there a characterization of knots whose mosaic number is realized only in non-minimal crossing representations?
• Is there an infinite family of knots whose mosaic number is realized only when the crossing number is not?

Bounds for Infinite Families

Current research has established bounds on mosaic numbers for specific infinite families of knots:

• L. & Wu (2012) showed that for torus knots T(p,p+1), m(T(p,p+1)) ≤ 2p
• H.J. Lee, K. Hong, H. Lee, and S. Oh showed that for torus knots T(p,q), m(T(p,q)) ≤ p+q-2 when |p-q| ≠ 1

Open questions include:

• Are these bounds tight?
• What are the mosaic numbers for other infinite families of knots, such as pretzel knots or Montesinos knots?
• Can we establish general formulas for the mosaic number of parameterized families of knots?

6.3 Efficiency and Optimization

Space Efficiency

The concept of space-efficient knot mosaics, introduced by Heap and Knowles, raises several questions:

• What is the minimum number of non-blank tiles (tile number) needed to represent a given knot on a mosaic of fixed size?
• For a given knot, what is the relationship between mosaic number and tile number?
• Is there a systematic way to construct space-efficient representations for arbitrary knots?

Algorithmic Complexity

The computational aspects of knot mosaics present several open questions:

• What is the computational complexity of determining the mosaic number of a knot?
• Can efficient algorithms be developed to find optimal knot mosaics for arbitrary knots?
• Are there heuristic approaches that can approximate optimal solutions for large or complex knots?

6.4 Structural Properties

Relationship to Other Knot Invariants

The relationship between mosaic number and other knot invariants remains an active area of research:

• What is the relationship between mosaic number and unknotting number?
• How does mosaic number relate to bridge number, arc index, or braid index?
• Can mosaic number be used to distinguish knots that other invariants cannot?

Grid Diagrams and Arc Presentations

The connection between knot mosaics, grid diagrams, and arc presentations offers several avenues for research:

• What is the precise relationship between knot mosaics, grid diagrams, and arc presentations?
• Ludwig's presentation notes that Arc index (K) = Grid Index (K), but how does this relate to mosaic number?
• Can results from grid diagram theory be translated to knot mosaics and vice versa?

6.5 Extensions and Generalizations

Corner Connection Tiles

The recent introduction of corner connection tiles by Heap et al. opens up new questions:

• How do corner connection tiles affect the mosaic number and tile number of knots?
• Is there a systematic way to convert between traditional mosaic representations and corner connection tile representations?
• Are there other alternative tile sets that might be more efficient for certain classes of knots?

3D Mosaic Extensions

The extension of knot mosaics to three dimensions presents intriguing possibilities:

• Can the knot mosaic system be meaningfully extended to three dimensions?
• What would be the appropriate tiles and moves for such a system?
• How would 3D mosaic numbers relate to other 3D knot invariants?

Virtual Knot Mosaics

The application of the mosaic approach to virtual knots is another potential area for research:

• How can the knot mosaic system be extended to virtual knots?
• What additional tiles would be needed, and how would this affect the mosaic number?
• What are the relationships between virtual knot invariants and virtual mosaic numbers?

6.6 Quantum Aspects

Quantum Entanglement and Topological Entanglement

The relationship between quantum and topological entanglement in quantum knot systems raises fascinating questions:

• What is the relationship between quantum entanglement and topological entanglement in quantum knot systems?
• Can quantum knots exhibit properties that classical knots cannot?
• How might quantum knot states be used in quantum information processing?

Quantum Tunneling

The quantum behavior of knot crossings presents another area for exploration:

• What is the nature of quantum tunneling of overcrossings into undercrossings in quantum knot systems?
• How does this affect the stability of quantum knots?
• Can this tunneling be observed or utilized in physical quantum systems?

7. Resources for Further Study

7.1 Online Resources

SUNY Geneseo Knot Mosaic Space

The Knot Mosaic Space website, created by Aaron Heap and his students at SUNY Geneseo, is an invaluable resource for anyone studying knot mosaics:

• Website: https://www.geneseo.edu/knotmosaics
• Features a complete table of knot mosaics for prime knots with crossing number 10 or less
• Provides an interactive Knot Mosaic Maker tool for creating and identifying knot mosaics
• Offers access to source code for pipeline search algorithms

arXiv Preprints

Many important papers on knot mosaics are available as preprints on arXiv:

• Lomonaco and Kauffman's original paper: arXiv:0805.0339
• Lee, Ludwig, Paat, and Peiffer's tabulation paper: arXiv:1602.03733
• Heap et al.'s paper on tabulating knots with crossing number 10 or less: arXiv:2303.12138
• Heap et al.'s paper on corner connection tiles: arXiv:2306.09276

7.2 Key Publications

Foundational Papers

• Lomonaco, S.J., Kauffman, L.H. (2008). Quantum Knots and Mosaics. Quantum Information Processing, 7, 85-115.
• Kuriya, T., Shehab, O. (2014). The Lomonaco-Kauffman Conjecture. Journal of Knot Theory and Its Ramifications, 23(1).

Mosaic Number and Crossing Number

• Lee, H.J., Hong, K., Lee, H., Oh, S. (2014). Mosaic Number of Knots. Journal of Knot Theory and Its Ramifications, 23(13).
• Ludwig, L., Evans, E., Paat, J. (2013). An Infinite Family of Knots Whose Mosaic Number Is Realized in Non-reduced Projections. Journal of Knot Theory and Its Ramifications, 22(7).

Tabulation and Space Efficiency

• Lee, H., Ludwig, L., Paat, J., Peiffer, A. (2018). Knot Mosaic Tabulation. Involve, 11(1), 13-26.
• Heap, A., Knowles, D. (2018). Tile Number and Space-Efficient Knot Mosaics. Journal of Knot Theory and Its Ramifications, 27(6).
• Heap, A., Knowles, D. (2019). Space-Efficient Knot Mosaics for Prime Knots with Mosaic Number 6. Involve, 12(5).
• Heap, A., LaCourt, N. (2020). Space-Efficient Prime Knot 7-Mosaics. Symmetry, 12(4).

Recent Developments

• Heap, A., Baldwin, D., Canning, J., Vinal, G. (2023). Tabulating Knot Mosaics: Crossing Number 10 or Less. Involve, 18, 91-104.
• Heap, A., et al. (2023). Knot Mosaics with Corner Connection Tiles. Pi Mu Epsilon Journal, 15(9), 553-568.

7.3 Books and Reference Materials

Knot Theory Textbooks

• Adams, C. (2004). The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. American Mathematical Society.
• Kauffman, L.H. (2001). Knots and Physics (3rd ed.). World Scientific.
• Lickorish, W.B.R. (1997). An Introduction to Knot Theory. Springer.

Encyclopedia and Reference Works

• Adams, C., Flapan, E., Henrich, A., Kauffman, L., Ludwig, L., Nelson, S. (2021). Encyclopedia of Knot Theory. Chapman and Hall/CRC.
  • This encyclopedia includes sections on knot mosaics and is a valuable reference for understanding the broader context of knot theory.

7.4 Software and Computational Tools

Knot Mosaic Maker

The interactive Knot Mosaic Maker tool on the SUNY Geneseo website allows users to:

• Create their own knot mosaics using a graphical interface
• Identify knots represented by mosaics
• Experiment with different mosaic representations of the same knot

Pipeline Search Algorithms

The source code for pipeline search algorithms used in tabulating knot mosaics is available through the SUNY Geneseo website, providing a starting point for computational research in this area.

7.5 Conferences and Workshops

Knot theory conferences often include presentations on knot mosaics. Some relevant conferences include:

• International Conference on Knot Theory and Its Ramifications
• AMS Special Sessions on Knot Theory
• Undergraduate mathematics conferences, which often feature accessible presentations on knot mosaics

7.6 Undergraduate Research Opportunities

Knot mosaics is an area particularly well-suited for undergraduate research, with opportunities including:

• Summer research programs, such as the one you'll be participating in with Professor Ludwig
• Research projects through the SUNY Geneseo Knot Mosaic Space
• Undergraduate research journals like Involve, which has published several papers on knot mosaics by undergraduate researchers

8. Conclusion

Knot mosaics represent a fascinating intersection of knot theory, combinatorics, and quantum physics. Since their introduction by Lomonaco and Kauffman in 2008, they have evolved from a theoretical framework for quantum knot systems into a vibrant area of mathematical research in their own right.

The field has seen significant development over the past fifteen years, with researchers establishing fundamental results about the relationship between knot mosaics and traditional knot theory, exploring the connections between mosaic number and other knot invariants, tabulating mosaic numbers for various classes of knots, and introducing innovations like space-efficient representations and corner connection tiles.

For an undergraduate student preparing to work with Professor Lew Ludwig on knot mosaics research, this field offers numerous opportunities for meaningful contributions. The discrete, combinatorial nature of knot mosaics makes them particularly accessible to undergraduate researchers, while the connections to deeper mathematical concepts and potential applications in quantum physics provide ample room for growth and exploration.

Professor Ludwig's work has been instrumental in advancing the field, particularly through his research on non-reduced projections and his contributions to the tabulation of mosaic numbers. His focus on making the field accessible to undergraduate researchers aligns perfectly with the nature of your upcoming summer research project.

As you prepare for this research experience, consider the many open questions that remain in knot mosaics research. Whether you're interested in the theoretical aspects of the relationship between mosaic number and other invariants, the computational challenges of finding optimal representations, or the innovative possibilities of new tile sets or extensions to other types of knots, there are abundant opportunities to make your own contributions to this evolving field.

The resources outlined in this survey, particularly the SUNY Geneseo Knot Mosaic Space website and the key publications in the field, will provide valuable starting points for your research. The interactive tools available online will allow you to experiment with knot mosaics and develop your intuition about their properties.

Knot mosaics represent a perfect blend of visual intuition and mathematical rigor, making them an ideal subject for undergraduate research. As you embark on your summer project with Professor Ludwig, you'll be joining a community of researchers who have found in knot mosaics a fascinating new perspective on one of mathematics' oldest and most intriguing subjects.

This survey has aimed to provide you with a comprehensive introduction to the field, from its foundational concepts to its current frontiers. As you delve deeper into specific aspects of knot mosaics during your research, you'll undoubtedly discover new connections, challenges, and insights that will contribute to your mathematical development and potentially to the field itself.

The study of knot mosaics continues to evolve, with new questions emerging as others are resolved. Your participation in this research area comes at an exciting time, with many fundamental questions still open and new approaches still being developed. We hope this survey serves as a valuable resource as you begin your exploration of knot mosaics with Professor Ludwig.

9. References

• Adams, C., Flapan, E., Henrich, A., Kauffman, L., Ludwig, L., & Nelson, S. (2021). Encyclopedia of Knot Theory. Chapman and Hall/CRC.
• Heap, A., & Knowles, D. (2018). Tile number and space-efficient knot mosaics. Journal of Knot Theory and Its Ramifications, 27(6).
• Heap, A., & Knowles, D. (2019). Space-efficient knot mosaics for prime knots with mosaic number 6. Involve, 12(5), 767-789.
• Heap, A., & LaCourt, N. (2020). Space-efficient prime knot 7-mosaics. Symmetry, 12(4), 576.
• Heap, A., Baldwin, D., Canning, J., & Vinal, G. (2023). Tabulating knot mosaics: Crossing number 10 or less. Involve, 18, 91-104.
• Heap, A., Donovan, U., Grossman, R., Laine, N., McDermott, C., Paone, M., & Southcott, D. (2023). Knot mosaics with corner connection tiles. Pi Mu Epsilon Journal, 15(9), 553-568.
• Kuriya, T., & Shehab, O. (2014). The Lomonaco-Kauffman conjecture. Journal of Knot Theory and Its Ramifications, 23(1).
• Lee, H. J., Hong, K., Lee, H., & Oh, S. (2014). Mosaic number of knots. Journal of Knot Theory and Its Ramifications, 23(13).
• Lee, H., Ludwig, L., Paat, J., & Peiffer, A. (2018). Knot mosaic tabulation. Involve, 11(1), 13-26.
• Lomonaco, S. J., & Kauffman, L. H. (2008). Quantum knots and mosaics. Quantum Information Processing, 7, 85-115.
• Ludwig, L., Evans, E., & Paat, J. (2013). An infinite family of knots whose mosaic number is realized in non-reduced projections. Journal of Knot Theory and Its Ramifications, 22(7).
• SUNY Geneseo. (n.d.). Knot Mosaic Space. Retrieved April 22, 2025, from https://www.geneseo.edu/knotmosaics