david freund, sarah smith-polderman, danielle shepherd...
TRANSCRIPT
Compiled from presentations and work by: David Freund, Sarah Smith-Polderman, Danielle Shepherd,
Joseph Smith, Michael Bush, and Katelyn French Advised by:
Dr. John Ramsay and Dr. Jennifer Bowen
Links are commonly represented as two-dimensional images called projections.
Knot projections are different representations of a given knot
Three projections of the trefoil knot [II].
In the mirror image of a link, every overcrossing is switched an undercrossing and vice versa
Most knots, like the trefoil knot, are not amphichiral (equivalent to their mirror image)
The left handed trefoil (left) and its mirror image, the right handed trefoil (right). There is no combination of Reidemeister
moves that can transform one into the other [IV].
Knots and links can be classified through their invariants
An invariant is an inherent characteristic of a link that is the same for any projection
The Perko Pair, they were thought to be different knots for 75 years [V]!
Crossing number P-colorability Linking number Knot groups Genus Bridge number Polynomials Braid Index
The crossing number of a knot or link A, denoted c(A), is the fewest number of crossings that occurs in any projection of the knot or link.
The crossing number of the trefoil is 3. The two projections on the left are at the trefoil’s crossing number, while the one on the right has 4 crossings [2].
*2
14Crossing number
Index (as catalogued by Rolfsen) [4]
Number of components
An asterisk denotes the mirror image of a link
2
14*2
14
Links organized by their Alexander-Briggs notations [VI].
There is only one knot with no crossings: the unknot
Millions of links have been catalogued
Add the values of all the crossings between different components
Divide the absolute value by 2
+1 1
A strand is a portion of a knot or link between two undercrossings.
A link is tricolorable if:
The three strands coming together at a crossing are all different colors or all the same color
At least two colors must be used to color the link
A tricolored trefoil knot [VII].
In a torus link, denoted T(m,n), the link wraps around the longitude m times and the meridian n times.
A link on a trefoil [VIII].
Link after Klein bottle has been removed Untangle the resulting link and classify
Digital Catalogue
87
3*
For an n-string braid B, 𝐵 = 𝐵𝜎𝑛 = 𝐵𝜎𝑛
−1.
Stabilization
𝐵 = 𝜎𝑖𝐵𝜎𝑖−1 = 𝜎𝑖
−1𝐵𝜎𝑖, where 1 ≤ 𝑖 ≤ 𝑛.
Conjugation
A general braid word for a torus link is [1]:
A general braid word for a Klein link adds a half-twist to the torus braid word:
m
n )( 121
)()( 11
2
1
1
1
1
121
inn
n
i
m
n
In a full twist (denoted Δ2), each string returns to its original position in the braid
In a half twist (denoted Δ), strings reverse their position in the braid
A full twist on a 5-string braid. A half twist on a 5-string braid.
When m ≥ n the general braid word of a Klein link can be simplified to the braid word w:
K(7,4)
Half Twist (Δ)
I. http://en.wikipedia.org/wiki/Figure-eight_knot_(mathematics)
II. http://vismath7.tripod.com/nat/#f4 III. http://mathworld.wolfram.com/ReidemeisterMoves.ht
ml IV. http://en.wikipedia.org/wiki/Trefoil_knot V. http://www.math.toronto.edu/drorbn/classes/0102/Feyn
manDiagrams/NonObvious/Perko_640.jpg VI. http://www.math.buffalo.edu/~menasco/knot-
theory.html VII. http://en.wikipedia.org/wiki/Tricolorability VIII. http://www.cgtp.duke.edu/~psa/cls/261/knot53.html IX. http://en.wikipedia.org/wiki/Klein_bottle X. http://www.wallpaperpin.com/wallpaper/1600x1200/rop
e-macro-knot-node-desktop-free-wallpaper-2458.html
1) Colin C. Adams. The Knot Book. American Mathematical Society, 2004.
2) W.B. Raymond Lickorish. An Introduction to Knot Theory. Springer, New York, 1997.
3) Kunio Murasugi. Knot Theory & Its Applications. Birkhäuser, 2008.
4) Dale Rolfsen. Knots and Links. Publish or Perish, Houston, 1976.