### Abstract

Since the Jones polynomial was discovered, the connection between knot theory and quantum physics has been of great interest. Lomonaco and Kauffman introduced the knot mosaic system to give a definition of the quantum knot system that is intended to represent an actual physical quantum system. Recently the authors developed an algorithm producing the exact enumeration of knot mosaics, which uses a recursion formula of state matrices. As a sequel to this research program, we similarly define the (embedded) graph mosaic system by using 16 graph mosaic tiles, representing graph diagrams with vertices of valence 3 and 4. We extend the algorithm to produce the exact number of all graph mosaics. The magnified state matrix that is an extension of the state matrix is mainly used.

Original language | English |
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Journal | Journal of Knot Theory and its Ramifications |

DOIs | |

Publication status | Accepted/In press - 2017 |

### Fingerprint

### Keywords

- graph mosaic
- knot mosaic
- Quantum knot

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Journal of Knot Theory and its Ramifications*. https://doi.org/10.1142/S0218216517500328

**Enumeration on graph mosaics.** / Hong, Kyungpyo; Oh, Seung Sang.

Research output: Contribution to journal › Article

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TY - JOUR

T1 - Enumeration on graph mosaics

AU - Hong, Kyungpyo

AU - Oh, Seung Sang

PY - 2017

Y1 - 2017

N2 - Since the Jones polynomial was discovered, the connection between knot theory and quantum physics has been of great interest. Lomonaco and Kauffman introduced the knot mosaic system to give a definition of the quantum knot system that is intended to represent an actual physical quantum system. Recently the authors developed an algorithm producing the exact enumeration of knot mosaics, which uses a recursion formula of state matrices. As a sequel to this research program, we similarly define the (embedded) graph mosaic system by using 16 graph mosaic tiles, representing graph diagrams with vertices of valence 3 and 4. We extend the algorithm to produce the exact number of all graph mosaics. The magnified state matrix that is an extension of the state matrix is mainly used.

AB - Since the Jones polynomial was discovered, the connection between knot theory and quantum physics has been of great interest. Lomonaco and Kauffman introduced the knot mosaic system to give a definition of the quantum knot system that is intended to represent an actual physical quantum system. Recently the authors developed an algorithm producing the exact enumeration of knot mosaics, which uses a recursion formula of state matrices. As a sequel to this research program, we similarly define the (embedded) graph mosaic system by using 16 graph mosaic tiles, representing graph diagrams with vertices of valence 3 and 4. We extend the algorithm to produce the exact number of all graph mosaics. The magnified state matrix that is an extension of the state matrix is mainly used.

KW - graph mosaic

KW - knot mosaic

KW - Quantum knot

UR - http://www.scopus.com/inward/record.url?scp=85015646727&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85015646727&partnerID=8YFLogxK

U2 - 10.1142/S0218216517500328

DO - 10.1142/S0218216517500328

M3 - Article

AN - SCOPUS:85015646727

JO - Journal of Knot Theory and its Ramifications

JF - Journal of Knot Theory and its Ramifications

SN - 0218-2165

ER -