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Introduction Hextile Number Conclusion
Introduction Hextile Number Conclusion
Hexagonal Mosaic Knots Malachi Alexander, Selina Foster, and Gianni - - PowerPoint PPT Presentation
Hexagonal Mosaic Knots Malachi Alexander, Selina Foster, and Gianni - - PowerPoint PPT Presentation
Introduction Hextile Number Conclusion Hexagonal Mosaic Knots Malachi Alexander, Selina Foster, and Gianni Krakoff Mentored by Dr. Jennifer McLoud-Mann University of Washington Bothell Research Experience for Undergraduates August 28, 2017
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Introduction Hextile Number Conclusion
Definition of a Knot
Definition A knot is an embedding of a circle in three-dimensional space.
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Definition of a Knot
Definition A knot is an embedding of a circle in three-dimensional space.
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Definition of a Knot
Definition A knot is an embedding of a circle in three-dimensional space.
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Introduction Hextile Number Conclusion
Common Terminology
Knot Crossing Unknot Undercrossing Link Overcrossing
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Introduction Hextile Number Conclusion
Equivalence of Knots
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Introduction Hextile Number Conclusion
Equivalence of Knots
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Introduction Hextile Number Conclusion
What is a Mosaic Knot?
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What is a Mosaic Knot?
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Introduction Hextile Number Conclusion
Defining the Hextile
A hextile must obey the following axioms: A curve must terminate at the midpoint of an edge and a curve cannot cross itself. Two curves cannot cross more than once and cannot share an edge. Examples of Violations
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Introduction Hextile Number Conclusion
Types of Hextiles
Type 0 Type 1 Type 2 Type 3 Type 4 Type 5 Type 6 Type 7 Type 8 Type 9 Type 10 Type 11 Type 12 Type 13 Type 14 Type 15 Type 16 Type 17 Type 18 Type 19 Type 21 Type 22 Type 23 Type 20
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Arrangements and Diagrams
Arrangement of ten hextiles. Arrangement of ten hextiles.
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Arrangements and Diagrams
Not suitably connected. Arrangement of ten hextiles.
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Arrangements and Diagrams
Not suitably connected. Suitably connected.
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Introduction Hextile Number Conclusion
Reidemeister Moves & Planar Isotopy
Reidemeister Move I Reidemeister Move II Reidemeister Move III Planar Isotopy
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Introduction Hextile Number Conclusion
The Hextile Number
Definition The hextile number of a link L is the least number of hextiles needed to represent L, denoted h(L).
More crossings per tile does not imply hextile number.
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Hextile Number as a Knot Invariant
Definition The hextile number of a link L is the least number of hextiles needed to represent L, denoted h(L). Theorem The hextile number is knot invariant. Proof. Similar to the crossing number, given two knots if the hextile numbers are different then the knots must be different, and if the hextile numbers are the same then we can’t conclude the knots are different.
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Introduction Hextile Number Conclusion
Theorems About Hextile Number
Theorem For a non-trivial link L, h(L) 6.
h(22
1) = 6 and h(31) = 6.
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Introduction Hextile Number Conclusion
Proof Concept: Pincer Movement
Construction To show that we actually have in our hands the hextile number for some knot, our main technique has been squeezing the upper and lower bounds. We want to show that h(31) = 6; it is sufficient to show that h(31) > 5, and that 7 > h(31). We already have it on 6. Computations become exponentially harder as the number of hextiles increases.
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Introduction Hextile Number Conclusion
Theorems About Hextile Number and Crossing Number
Theorem For a link L, if c(L) 4, then h(L) 8. Theorem For a knot K, if c(K) 5, then h(K) 9.
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Known and Unknown Hextile Numbers
L h(L) 01 3 02
1
5 22
1
6 31 6 41 8 42
1
8 52
1
8 51 9 52 9 31#31 9
Is this D(73) reducible?
L h(L) 61 9 62 9 63 9 71 12? 72 13? 73 14? 74 11? 75 11? 76 10? 77 9
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Introduction Hextile Number Conclusion
Thank You!
I would like to thank: My research mentor, Dr. Jennifer McLoud-Mann My faculty mentor, Dr. Alison Lynch The National Science Foundation Grant DMS1460699 Undergraduate Research Opportunities Center
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