L11n162

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L11n161.gif

L11n161

L11n163.gif

L11n163

Contents

L11n162.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11n162's Link Presentations]

Planar diagram presentation X8192 X10,3,11,4 X17,13,18,12 X14,5,15,6 X4,13,5,14 X11,19,12,18 X19,7,20,22 X15,21,16,20 X21,17,22,16 X2738 X6,9,1,10
Gauss code {1, -10, 2, -5, 4, -11}, {10, -1, 11, -2, -6, 3, 5, -4, -8, 9, -3, 6, -7, 8, -9, 7}
A Braid Representative
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A Morse Link Presentation L11n162 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{t(1) t(2)^4+t(2)^4+t(1)^2 t(2)^3-2 t(1) t(2)^3-t(1)^2 t(2)^2+3 t(1) t(2)^2-t(2)^2-2 t(1) t(2)+t(2)+t(1)^2+t(1)}{t(1) t(2)^2} (db)
Jones polynomial q^{9/2}-2 q^{7/2}+3 q^{5/2}-\frac{2}{q^{5/2}}-4 q^{3/2}+\frac{3}{q^{3/2}}-\frac{1}{q^{11/2}}+4 \sqrt{q}-\frac{4}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial z a^5+a^5 z^{-1} -z^5 a-4 z^3 a-5 z a-2 a z^{-1} -z^5 a^{-1} -3 z^3 a^{-1} -z a^{-1} + a^{-1} z^{-1} +z^3 a^{-3} +2 z a^{-3} (db)
Kauffman polynomial a^5 z^7-7 a^5 z^5+14 a^5 z^3-9 a^5 z+a^5 z^{-1} +z^6 a^{-4} -2 a^4 z^4-4 z^4 a^{-4} +4 a^4 z^2+3 z^2 a^{-4} -a^4+2 z^7 a^{-3} -2 a^3 z^5-8 z^5 a^{-3} +6 a^3 z^3+7 z^3 a^{-3} -3 a^3 z-2 z a^{-3} +a^2 z^8+2 z^8 a^{-2} -7 a^2 z^6-8 z^6 a^{-2} +15 a^2 z^4+8 z^4 a^{-2} -10 a^2 z^2-5 z^2 a^{-2} +3 a^2+2 a^{-2} +a z^9+z^9 a^{-1} -6 a z^7-3 z^7 a^{-1} +13 a z^5-13 a z^3+2 z^3 a^{-1} +8 a z-2 a z^{-1} - a^{-1} z^{-1} +3 z^8-16 z^6+29 z^4-22 z^2+5 (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 \
-6-5-4-3-2-1012345χ
10           1-1
8          1 1
6         21 -1
4        21  1
2       22   0
0     132    0
-2     23     1
-4   122      -1
-6    2       2
-8  11        0
-101           1
-121           1
Integral Khovanov Homology

(db, data source)

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

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