L11n302

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L11n301.gif

L11n301

L11n303.gif

L11n303

Contents

L11n302.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n302 at Knotilus!


Link Presentations

[edit Notes on L11n302's Link Presentations]

Planar diagram presentation X6172 X3,13,4,12 X20,13,21,14 X22,19,11,20 X15,5,16,10 X17,9,18,8 X7,17,8,16 X9,19,10,18 X14,21,15,22 X2536 X11,1,12,4
Gauss code {1, -10, -2, 11}, {10, -1, -7, 6, -8, 5}, {-11, 2, 3, -9, -5, 7, -6, 8, 4, -3, 9, -4}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart1.gif
BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11n302 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{-u v^2 w+u v w-u v-u w^2+u w-v^2 w^3+v^2 w^2+v w^4-v w^3+w^3}{\sqrt{u} v w^2} (db)
Jones polynomial q^6-q^5+2 q^4-q^3+2 q^2+1+ q^{-1} - q^{-2} + q^{-3} - q^{-4} (db)
Signature 0 (db)
HOMFLY-PT polynomial -z^6 a^{-2} +z^6-a^2 z^4-7 z^4 a^{-2} +z^4 a^{-4} +7 z^4-4 a^2 z^2-17 z^2 a^{-2} +4 z^2 a^{-4} +16 z^2-4 a^2-16 a^{-2} +5 a^{-4} +15-a^2 z^{-2} -5 a^{-2} z^{-2} +2 a^{-4} z^{-2} +4 z^{-2} (db)
Kauffman polynomial a^2 z^8+z^8 a^{-2} +z^8 a^{-4} +z^8+a^3 z^7+2 a z^7+2 z^7 a^{-1} +2 z^7 a^{-3} +z^7 a^{-5} -6 a^2 z^6-8 z^6 a^{-2} -5 z^6 a^{-4} +z^6 a^{-6} -8 z^6-6 a^3 z^5-14 a z^5-16 z^5 a^{-1} -12 z^5 a^{-3} -4 z^5 a^{-5} +10 a^2 z^4+22 z^4 a^{-2} +7 z^4 a^{-4} -5 z^4 a^{-6} +20 z^4+10 a^3 z^3+28 a z^3+37 z^3 a^{-1} +21 z^3 a^{-3} +2 z^3 a^{-5} -7 a^2 z^2-31 z^2 a^{-2} -7 z^2 a^{-4} +6 z^2 a^{-6} -25 z^2-5 a^3 z-21 a z-33 z a^{-1} -16 z a^{-3} +z a^{-5} +4 a^2+20 a^{-2} +6 a^{-4} -2 a^{-6} +17+a^3 z^{-1} +5 a z^{-1} +9 a^{-1} z^{-1} +5 a^{-3} z^{-1} -a^2 z^{-2} -5 a^{-2} z^{-2} -2 a^{-4} z^{-2} -4 z^{-2} (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 \
-5-4-3-2-10123456χ
13           11
11          110
9         1  1
7       111  1
5      131   1
3     1 1    2
1    142     1
-1   1 1      2
-3   11       0
-5 11         0
-7            0
-91           -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1 i=3
r=-5 {\mathbb Z}
r=-4 {\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}
r=-2 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z} {\mathbb Z}_2 {\mathbb Z}
r=0 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}^{4} {\mathbb Z}
r=1 {\mathbb Z}^{2} {\mathbb Z}_2 {\mathbb Z}
r=2 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3} {\mathbb Z}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=5 {\mathbb Z}_2 {\mathbb Z}
r=6 {\mathbb Z} {\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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See/edit the Link Page master template (intermediate).

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L11n303