L11n74

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L11n73.gif

L11n73

L11n75.gif

L11n75

Contents

L11n74.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11n74's Link Presentations]

Planar diagram presentation X6172 X10,3,11,4 X11,19,12,18 X7,17,8,16 X17,9,18,8 X13,21,14,20 X15,5,16,22 X19,13,20,12 X21,15,22,14 X2536 X4,9,1,10
Gauss code {1, -10, 2, -11}, {10, -1, -4, 5, 11, -2, -3, 8, -6, 9, -7, 4, -5, 3, -8, 6, -9, 7}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart3.gif
BraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart1.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart2.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11n74 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{-u v^5+u v^4-u v^3+u v^2-u-v^7+v^5-v^4+v^3-v^2}{\sqrt{u} v^{7/2}} (db)
Jones polynomial 3 q^{9/2}-2 q^{7/2}+q^{5/2}-q^{3/2}-\frac{1}{q^{3/2}}+q^{17/2}-q^{15/2}+2 q^{13/2}-3 q^{11/2}-\sqrt{q} (db)
Signature 3 (db)
HOMFLY-PT polynomial z^3 a^{-7} +2 z a^{-7} +2 z^3 a^{-5} +5 z a^{-5} +3 a^{-5} z^{-1} -z^7 a^{-3} -8 z^5 a^{-3} -20 z^3 a^{-3} -19 z a^{-3} -7 a^{-3} z^{-1} +z^5 a^{-1} +6 z^3 a^{-1} +10 z a^{-1} +4 a^{-1} z^{-1} (db)
Kauffman polynomial z^4 a^{-10} -3 z^2 a^{-10} + a^{-10} +z^5 a^{-9} -2 z^3 a^{-9} +z^6 a^{-8} -2 z^4 a^{-8} +z^2 a^{-8} +z^7 a^{-7} -4 z^5 a^{-7} +7 z^3 a^{-7} -2 z a^{-7} +z^6 a^{-6} -2 z^4 a^{-6} +3 z^2 a^{-6} +z^5 a^{-5} -3 z^3 a^{-5} +7 z a^{-5} -3 a^{-5} z^{-1} +z^8 a^{-4} -9 z^6 a^{-4} +26 z^4 a^{-4} -26 z^2 a^{-4} +7 a^{-4} +z^9 a^{-3} -10 z^7 a^{-3} +34 z^5 a^{-3} -49 z^3 a^{-3} +30 z a^{-3} -7 a^{-3} z^{-1} +z^8 a^{-2} -9 z^6 a^{-2} +25 z^4 a^{-2} -25 z^2 a^{-2} +7 a^{-2} +z^9 a^{-1} -9 z^7 a^{-1} +28 z^5 a^{-1} -37 z^3 a^{-1} +21 z a^{-1} -4 a^{-1} z^{-1} (db)

Khovanov Homology

The coefficients of the monomials t^rq^j are shown, along with their alternating sums \chi (fixed j, alternation over r).   
\ r
  \  
j \
-4-3-2-101234567χ
18           1-1
16            0
14         21 -1
12        1   1
10       22   0
8     111    -1
6     12     1
4   121      0
2    2       2
0  1         1
-21           1
-41           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=2 i=4 i=6
r=-4 {\mathbb Z} {\mathbb Z}
r=-3
r=-2 {\mathbb Z}
r=-1 {\mathbb Z}_2 {\mathbb Z}
r=0 {\mathbb Z}^{2} {\mathbb Z}^{2}
r=1 {\mathbb Z} {\mathbb Z} {\mathbb Z}
r=2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=5 {\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=6 {\mathbb Z}
r=7 {\mathbb Z}_2 {\mathbb Z}

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

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L11n73.gif

L11n73

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L11n75