L11a122

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L11a121.gif

L11a121

L11a123.gif

L11a123

Contents

L11a122.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a122 at Knotilus!


Link Presentations

[edit Notes on L11a122's Link Presentations]

Planar diagram presentation X6172 X14,3,15,4 X18,8,19,7 X22,20,5,19 X20,9,21,10 X8,21,9,22 X16,12,17,11 X12,16,13,15 X10,18,11,17 X2536 X4,13,1,14
Gauss code {1, -10, 2, -11}, {10, -1, 3, -6, 5, -9, 7, -8, 11, -2, 8, -7, 9, -3, 4, -5, 6, -4}
A Braid Representative
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A Morse Link Presentation L11a122 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{2 u v^3-8 u v^2+10 u v-3 u-3 v^3+10 v^2-8 v+2}{\sqrt{u} v^{3/2}} (db)
Jones polynomial q^{9/2}-4 q^{7/2}+7 q^{5/2}-10 q^{3/2}+13 \sqrt{q}-\frac{15}{\sqrt{q}}+\frac{14}{q^{3/2}}-\frac{12}{q^{5/2}}+\frac{8}{q^{7/2}}-\frac{5}{q^{9/2}}+\frac{2}{q^{11/2}}-\frac{1}{q^{13/2}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a^7 z^{-1} -3 a^5 z-2 a^5 z^{-1} +3 a^3 z^3+z^3 a^{-3} +3 a^3 z+2 a^3 z^{-1} -a z^5-z^5 a^{-1} -z^3 a^{-1} -a z-a z^{-1} -z a^{-1} (db)
Kauffman polynomial a^7 z^5-3 a^7 z^3+3 a^7 z-a^7 z^{-1} +2 a^6 z^6-4 a^6 z^4+2 a^6 z^2+2 a^5 z^7+a^5 z^5-9 a^5 z^3+8 a^5 z-2 a^5 z^{-1} +2 a^4 z^8+2 a^4 z^6+z^6 a^{-4} -7 a^4 z^4-2 z^4 a^{-4} +4 a^4 z^2-a^4+2 a^3 z^9-a^3 z^7+4 z^7 a^{-3} +6 a^3 z^5-11 z^5 a^{-3} -13 a^3 z^3+5 z^3 a^{-3} +9 a^3 z-2 a^3 z^{-1} +a^2 z^{10}+3 a^2 z^8+6 z^8 a^{-2} -7 a^2 z^6-18 z^6 a^{-2} +8 a^2 z^4+13 z^4 a^{-2} -3 a^2 z^2-2 z^2 a^{-2} +6 a z^9+4 z^9 a^{-1} -14 a z^7-7 z^7 a^{-1} +14 a z^5-3 z^5 a^{-1} -6 a z^3+6 z^3 a^{-1} +3 a z-z a^{-1} -a z^{-1} +z^{10}+7 z^8-26 z^6+26 z^4-7 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         41 -3
4        63  3
2       74   -3
0      86    2
-2     78     1
-4    57      -2
-6   37       4
-8  25        -3
-10 14         3
-12 1          -1
-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}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=-3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=0 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{8}
r=1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
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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L11a121

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L11a123