L11n108

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L11n107.gif

L11n107

L11n109.gif

L11n109

Contents

L11n108.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11n108's Link Presentations]

Planar diagram presentation X6172 X12,3,13,4 X13,19,14,18 X17,11,18,10 X21,9,22,8 X7,17,8,16 X9,21,10,20 X15,5,16,22 X19,15,20,14 X2536 X4,11,1,12
Gauss code {1, -10, 2, -11}, {10, -1, -6, 5, -7, 4, 11, -2, -3, 9, -8, 6, -4, 3, -9, 7, -5, 8}
A Braid Representative
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A Morse Link Presentation L11n108 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{-u v^4+3 u v^3-2 u v^2-u v+2 u+2 v^5-v^4-2 v^3+3 v^2-v}{\sqrt{u} v^{5/2}} (db)
Jones polynomial -3 q^{9/2}+3 q^{7/2}-2 q^{5/2}-\frac{1}{q^{5/2}}+q^{3/2}+\frac{1}{q^{3/2}}+q^{15/2}-2 q^{13/2}+3 q^{11/2}-\sqrt{q}-\frac{2}{\sqrt{q}} (db)
Signature 1 (db)
HOMFLY-PT polynomial -z^5 a^{-1} +a z^3-5 z^3 a^{-1} -z^3 a^{-3} -z^3 a^{-5} +3 a z-5 z a^{-1} -z a^{-3} +z a^{-7} +2 a z^{-1} -2 a^{-1} z^{-1} - a^{-3} z^{-1} + a^{-5} z^{-1} (db)
Kauffman polynomial -z^9 a^{-1} -z^9 a^{-3} -2 z^8 a^{-2} -2 z^8 a^{-4} -z^8 a^{-6} -z^8-a z^7+6 z^7 a^{-1} +6 z^7 a^{-3} -3 z^7 a^{-5} -2 z^7 a^{-7} +14 z^6 a^{-2} +12 z^6 a^{-4} +2 z^6 a^{-6} -z^6 a^{-8} +5 z^6+6 a z^5-7 z^5 a^{-1} -8 z^5 a^{-3} +13 z^5 a^{-5} +8 z^5 a^{-7} -24 z^4 a^{-2} -21 z^4 a^{-4} +4 z^4 a^{-6} +4 z^4 a^{-8} -3 z^4-10 a z^3-2 z^3 a^{-1} +3 z^3 a^{-3} -12 z^3 a^{-5} -7 z^3 a^{-7} +10 z^2 a^{-2} +15 z^2 a^{-4} -3 z^2 a^{-6} -4 z^2 a^{-8} -4 z^2+7 a z+4 z a^{-1} -3 z a^{-3} +2 z a^{-5} +2 z a^{-7} -3 a^{-4} + a^{-8} +3-2 a z^{-1} -2 a^{-1} z^{-1} + a^{-3} z^{-1} + a^{-5} 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
16           1-1
14          1 1
12         21 -1
10       121  0
8       22   0
6     232    -1
4    122     1
2   132      0
0  113       3
-2  1         1
-411          0
-61           1
Integral Khovanov Homology

(db, data source)

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

Read me first: Modifying Knot Pages

See/edit the Link Page master template (intermediate).

See/edit the Link_Splice_Base (expert).

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