L11a231

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L11a230.gif

L11a230

L11a232.gif

L11a232

Contents

L11a231.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a231 at Knotilus!


Link Presentations

[edit Notes on L11a231's Link Presentations]

Planar diagram presentation X8192 X12,3,13,4 X22,13,7,14 X14,9,15,10 X10,21,11,22 X18,5,19,6 X20,16,21,15 X16,20,17,19 X2738 X4,11,5,12 X6,17,1,18
Gauss code {1, -9, 2, -10, 6, -11}, {9, -1, 4, -5, 10, -2, 3, -4, 7, -8, 11, -6, 8, -7, 5, -3}
A Braid Representative
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A Morse Link Presentation L11a231 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{2 u^2 v^3-7 u^2 v^2+6 u^2 v-2 u^2+u v^4-8 u v^3+13 u v^2-8 u v+u-2 v^4+6 v^3-7 v^2+2 v}{u v^2} (db)
Jones polynomial \frac{21}{q^{9/2}}-\frac{19}{q^{7/2}}+\frac{14}{q^{5/2}}-\frac{9}{q^{3/2}}+\frac{1}{q^{21/2}}-\frac{3}{q^{19/2}}+\frac{7}{q^{17/2}}-\frac{13}{q^{15/2}}+\frac{17}{q^{13/2}}-\frac{21}{q^{11/2}}-\sqrt{q}+\frac{4}{\sqrt{q}} (db)
Signature -3 (db)
HOMFLY-PT polynomial -a^{11} z^{-1} +4 a^9 z+3 a^9 z^{-1} -5 a^7 z^3-6 a^7 z-2 a^7 z^{-1} +2 a^5 z^5+2 a^5 z^3+a^5 z+a^3 z^5-a^3 z^3-2 a^3 z-a z^3 (db)
Kauffman polynomial a^{12} z^6-3 a^{12} z^4+3 a^{12} z^2-a^{12}+3 a^{11} z^7-8 a^{11} z^5+7 a^{11} z^3-3 a^{11} z+a^{11} z^{-1} +4 a^{10} z^8-5 a^{10} z^6-6 a^{10} z^4+9 a^{10} z^2-3 a^{10}+4 a^9 z^9-2 a^9 z^7-11 a^9 z^5+15 a^9 z^3-12 a^9 z+3 a^9 z^{-1} +2 a^8 z^{10}+6 a^8 z^8-17 a^8 z^6+8 a^8 z^4+3 a^8 z^2-3 a^8+11 a^7 z^9-20 a^7 z^7+9 a^7 z^5+6 a^7 z^3-7 a^7 z+2 a^7 z^{-1} +2 a^6 z^{10}+12 a^6 z^8-34 a^6 z^6+31 a^6 z^4-8 a^6 z^2+7 a^5 z^9-7 a^5 z^7-2 a^5 z^5+5 a^5 z^3+10 a^4 z^8-19 a^4 z^6+15 a^4 z^4-5 a^4 z^2+8 a^3 z^7-13 a^3 z^5+6 a^3 z^3-2 a^3 z+4 a^2 z^6-5 a^2 z^4+a z^5-a z^3 (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 \
-9-8-7-6-5-4-3-2-1012χ
2           11
0          3 -3
-2         61 5
-4        94  -5
-6       105   5
-8      119    -2
-10     1010     0
-12    711      4
-14   610       -4
-16  28        6
-18 15         -4
-20 2          2
-221           -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-4 i=-2
r=-9 {\mathbb Z}
r=-8 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-7 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-6 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{6}
r=-5 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-4 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=-3 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=-2 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=-1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=0 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{6}
r=1 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=2 {\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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