L11a271

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L11a270.gif

L11a270

L11a272.gif

L11a272

Contents

L11a271.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11a271's Link Presentations]

Planar diagram presentation X10,1,11,2 X2,11,3,12 X12,3,13,4 X4,9,5,10 X14,5,15,6 X18,8,19,7 X22,15,9,16 X20,17,21,18 X16,21,17,22 X6,20,7,19 X8,13,1,14
Gauss code {1, -2, 3, -4, 5, -10, 6, -11}, {4, -1, 2, -3, 11, -5, 7, -9, 8, -6, 10, -8, 9, -7}
A Braid Representative
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A Morse Link Presentation L11a271 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{2 u^3 v^2-2 u^3 v+2 u^2 v^3-7 u^2 v^2+6 u^2 v-2 u^2-2 u v^3+6 u v^2-7 u v+2 u-2 v^2+2 v}{u^{3/2} v^{3/2}} (db)
Jones polynomial -\sqrt{q}+\frac{2}{\sqrt{q}}-\frac{5}{q^{3/2}}+\frac{8}{q^{5/2}}-\frac{11}{q^{7/2}}+\frac{13}{q^{9/2}}-\frac{14}{q^{11/2}}+\frac{11}{q^{13/2}}-\frac{9}{q^{15/2}}+\frac{6}{q^{17/2}}-\frac{3}{q^{19/2}}+\frac{1}{q^{21/2}} (db)
Signature -3 (db)
HOMFLY-PT polynomial a^9 \left(-z^3\right)-a^9 z+a^7 z^5+a^7 z^3-a^7 z+2 a^5 z^5+5 a^5 z^3+4 a^5 z+a^5 z^{-1} +a^3 z^5+a^3 z^3-2 a^3 z-a^3 z^{-1} -a z^3-2 a z (db)
Kauffman polynomial a^{12} z^6-3 a^{12} z^4+2 a^{12} z^2+3 a^{11} z^7-9 a^{11} z^5+6 a^{11} z^3-a^{11} z+4 a^{10} z^8-11 a^{10} z^6+7 a^{10} z^4-2 a^{10} z^2+3 a^9 z^9-6 a^9 z^7+3 a^9 z^5-4 a^9 z^3+2 a^9 z+a^8 z^{10}+3 a^8 z^8-11 a^8 z^6+9 a^8 z^4-3 a^8 z^2+5 a^7 z^9-12 a^7 z^7+13 a^7 z^5-6 a^7 z^3+3 a^7 z+a^6 z^{10}+2 a^6 z^8-6 a^6 z^6+8 a^6 z^4-2 a^6 z^2+2 a^5 z^9-6 a^5 z^5+12 a^5 z^3-6 a^5 z+a^5 z^{-1} +3 a^4 z^8-5 a^4 z^6+5 a^4 z^4-2 a^4 z^2-a^4+3 a^3 z^7-6 a^3 z^5+5 a^3 z^3-4 a^3 z+a^3 z^{-1} +2 a^2 z^6-4 a^2 z^4+a^2 z^2+a z^5-3 a z^3+2 a z (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          1 -1
-2         41 3
-4        52  -3
-6       63   3
-8      75    -2
-10     76     1
-12    58      3
-14   46       -2
-16  25        3
-18 14         -3
-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}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-6 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-5 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-4 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{7}
r=-3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=0 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{4}
r=1 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
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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L11a270

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L11a272