L11a236

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L11a235.gif

L11a235

L11a237.gif

L11a237

Contents

L11a236.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a236 at Knotilus!


Link Presentations

[edit Notes on L11a236's Link Presentations]

Planar diagram presentation X8192 X12,3,13,4 X20,15,21,16 X16,9,17,10 X10,19,11,20 X22,13,7,14 X14,21,15,22 X18,5,19,6 X2738 X4,11,5,12 X6,17,1,18
Gauss code {1, -9, 2, -10, 8, -11}, {9, -1, 4, -5, 10, -2, 6, -7, 3, -4, 11, -8, 5, -3, 7, -6}
A Braid Representative
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A Morse Link Presentation L11a236 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{5 u^2 v^2-6 u^2 v+2 u^2+5 u v^3-9 u v^2+5 u v+2 v^4-6 v^3+5 v^2}{u v^2} (db)
Jones polynomial -\frac{1}{q^{3/2}}+\frac{3}{q^{5/2}}-\frac{6}{q^{7/2}}+\frac{9}{q^{9/2}}-\frac{13}{q^{11/2}}+\frac{13}{q^{13/2}}-\frac{14}{q^{15/2}}+\frac{12}{q^{17/2}}-\frac{9}{q^{19/2}}+\frac{6}{q^{21/2}}-\frac{3}{q^{23/2}}+\frac{1}{q^{25/2}} (db)
Signature -3 (db)
HOMFLY-PT polynomial -a^{13} z^{-1} +4 a^{11} z+2 a^{11} z^{-1} -3 a^9 z^3-a^9 z-5 a^7 z^3-5 a^7 z-a^7 z^{-1} -3 a^5 z^3-a^5 z-a^3 z^3 (db)
Kauffman polynomial a^{14} z^8-5 a^{14} z^6+9 a^{14} z^4-7 a^{14} z^2+2 a^{14}+3 a^{13} z^9-15 a^{13} z^7+24 a^{13} z^5-13 a^{13} z^3+2 a^{13} z-a^{13} z^{-1} +2 a^{12} z^{10}-3 a^{12} z^8-18 a^{12} z^6+44 a^{12} z^4-29 a^{12} z^2+5 a^{12}+10 a^{11} z^9-42 a^{11} z^7+51 a^{11} z^5-21 a^{11} z^3+7 a^{11} z-2 a^{11} z^{-1} +2 a^{10} z^{10}+7 a^{10} z^8-50 a^{10} z^6+64 a^{10} z^4-25 a^{10} z^2+3 a^{10}+7 a^9 z^9-15 a^9 z^7-6 a^9 z^5+14 a^9 z^3-2 a^9 z+11 a^8 z^8-28 a^8 z^6+15 a^8 z^4-2 a^8 z^2-a^8+12 a^7 z^7-27 a^7 z^5+17 a^7 z^3-6 a^7 z+a^7 z^{-1} +9 a^6 z^6-11 a^6 z^4+a^6 z^2+6 a^5 z^5-4 a^5 z^3+a^5 z+3 a^4 z^4+a^3 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 \
-11-10-9-8-7-6-5-4-3-2-10χ
-2           11
-4          31-2
-6         3  3
-8        63  -3
-10       73   4
-12      66    0
-14     87     1
-16    57      2
-18   47       -3
-20  25        3
-22 14         -3
-24 2          2
-261           -1
Integral Khovanov Homology

(db, data source)

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

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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