L11a138

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L11a137.gif

L11a137

L11a139.gif

L11a139

Contents

L11a138.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11a138's Link Presentations]

Planar diagram presentation X6172 X2,11,3,12 X14,3,15,4 X12,5,13,6 X22,13,5,14 X4,21,1,22 X20,16,21,15 X16,8,17,7 X8,20,9,19 X18,10,19,9 X10,18,11,17
Gauss code {1, -2, 3, -6}, {4, -1, 8, -9, 10, -11, 2, -4, 5, -3, 7, -8, 11, -10, 9, -7, 6, -5}
A Braid Representative
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A Morse Link Presentation L11a138 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{4 t(1) t(2)^3-4 t(2)^3-12 t(1) t(2)^2+13 t(2)^2+13 t(1) t(2)-12 t(2)-4 t(1)+4}{\sqrt{t(1)} t(2)^{3/2}} (db)
Jones polynomial q^{9/2}-4 q^{7/2}+8 q^{5/2}-13 q^{3/2}+18 \sqrt{q}-\frac{21}{\sqrt{q}}+\frac{21}{q^{3/2}}-\frac{19}{q^{5/2}}+\frac{13}{q^{7/2}}-\frac{9}{q^{9/2}}+\frac{4}{q^{11/2}}-\frac{1}{q^{13/2}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a^5 z^3+a^5 z^{-1} -a^3 z^5+z^3 a^{-3} -a^3 z-a^3 z^{-1} -2 a z^5-z^5 a^{-1} -2 a z^3-a z (db)
Kauffman polynomial a^7 z^5-a^7 z^3+4 a^6 z^6-5 a^6 z^4+8 a^5 z^7-14 a^5 z^5+8 a^5 z^3-3 a^5 z+a^5 z^{-1} +9 a^4 z^8-15 a^4 z^6+z^6 a^{-4} +10 a^4 z^4-2 z^4 a^{-4} -3 a^4 z^2-a^4+6 a^3 z^9-2 a^3 z^7+4 z^7 a^{-3} -10 a^3 z^5-10 z^5 a^{-3} +9 a^3 z^3+5 z^3 a^{-3} -2 a^3 z+a^3 z^{-1} +2 a^2 z^{10}+12 a^2 z^8+7 z^8 a^{-2} -35 a^2 z^6-19 z^6 a^{-2} +34 a^2 z^4+15 z^4 a^{-2} -11 a^2 z^2-4 z^2 a^{-2} +12 a z^9+6 z^9 a^{-1} -25 a z^7-11 z^7 a^{-1} +18 a z^5+3 z^5 a^{-1} -4 a z^3+z^3 a^{-1} +a z+2 z^{10}+10 z^8-36 z^6+36 z^4-12 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 \
-6-5-4-3-2-1012345χ
10           1-1
8          3 3
6         51 -4
4        83  5
2       105   -5
0      118    3
-2     1111     0
-4    810      -2
-6   511       6
-8  48        -4
-10 16         5
-12 3          -3
-141           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-2 i=0
r=-6 {\mathbb Z}
r=-5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{4}
r=-3 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-2 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=-1 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=0 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{11}
r=1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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

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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L11a137

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L11a139