L11n236

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L11n235.gif

L11n235

L11n237.gif

L11n237

Contents

L11n236.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n236 at Knotilus!


Link Presentations

[edit Notes on L11n236's Link Presentations]

Planar diagram presentation X10,1,11,2 X2,11,3,12 X14,3,15,4 X20,13,21,14 X12,21,13,22 X22,5,9,6 X7,16,8,17 X15,18,16,19 X17,8,18,1 X6,9,7,10 X4,19,5,20
Gauss code {1, -2, 3, -11, 6, -10, -7, 9}, {10, -1, 2, -5, 4, -3, -8, 7, -9, 8, 11, -4, 5, -6}
A Braid Representative
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BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart4.gif
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A Morse Link Presentation L11n236 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{\left(2 u^2 v-u^2-v+2\right) \left(u v^2+1\right)}{u^{3/2} v^{3/2}} (db)
Jones polynomial -\frac{1}{q^{7/2}}+\frac{1}{q^{9/2}}-\frac{2}{q^{11/2}}+\frac{1}{q^{19/2}}-\frac{1}{q^{21/2}}+\frac{1}{q^{23/2}}-\frac{1}{q^{25/2}} (db)
Signature -6 (db)
HOMFLY-PT polynomial a^{13} (-z)+a^{11} z^5+6 a^{11} z^3+6 a^{11} z-a^9 z^7-6 a^9 z^5-9 a^9 z^3-3 a^9 z+a^9 z^{-1} -a^7 z^7-6 a^7 z^5-10 a^7 z^3-6 a^7 z-a^7 z^{-1} (db)
Kauffman polynomial -z^5 a^{15}+4 z^3 a^{15}-3 z a^{15}-z^6 a^{14}+4 z^4 a^{14}-2 z^2 a^{14}-z^5 a^{13}+5 z^3 a^{13}-3 z a^{13}-z^8 a^{12}+7 z^6 a^{12}-14 z^4 a^{12}+10 z^2 a^{12}-z^9 a^{11}+7 z^7 a^{11}-15 z^5 a^{11}+15 z^3 a^{11}-7 z a^{11}-2 z^8 a^{10}+13 z^6 a^{10}-22 z^4 a^{10}+10 z^2 a^{10}-z^9 a^9+6 z^7 a^9-9 z^5 a^9+4 z^3 a^9-z a^9-a^9 z^{-1} -z^8 a^8+5 z^6 a^8-4 z^4 a^8-2 z^2 a^8+a^8-z^7 a^7+6 z^5 a^7-10 z^3 a^7+6 z a^7-a^7 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 \
-10-9-8-7-6-5-4-3-2-10χ
-6          11
-8         110
-10        1  1
-12      211  2
-14     121   0
-16    121    0
-18   122     -1
-20  11       0
-22 121       0
-24           0
-261          1
Integral Khovanov Homology

(db, data source)

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

See/edit the Link_Splice_Base (expert).

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L11n235.gif

L11n235

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L11n237