L11a196

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L11a195.gif

L11a195

L11a197.gif

L11a197

Contents

L11a196.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a196 at Knotilus!


Link Presentations

[edit Notes on L11a196's Link Presentations]

Planar diagram presentation X8192 X10,4,11,3 X22,10,7,9 X2738 X20,13,21,14 X6,12,1,11 X18,15,19,16 X4,18,5,17 X16,6,17,5 X14,19,15,20 X12,21,13,22
Gauss code {1, -4, 2, -8, 9, -6}, {4, -1, 3, -2, 6, -11, 5, -10, 7, -9, 8, -7, 10, -5, 11, -3}
A Braid Representative
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A Morse Link Presentation L11a196 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{2 u^2 v^2-5 u^2 v+3 u^2-5 u v^2+11 u v-5 u+3 v^2-5 v+2}{u v} (db)
Jones polynomial q^{11/2}-3 q^{9/2}+5 q^{7/2}-9 q^{5/2}+11 q^{3/2}-13 \sqrt{q}+\frac{12}{\sqrt{q}}-\frac{11}{q^{3/2}}+\frac{8}{q^{5/2}}-\frac{5}{q^{7/2}}+\frac{3}{q^{9/2}}-\frac{1}{q^{11/2}} (db)
Signature 1 (db)
HOMFLY-PT polynomial a^5 z+z a^{-5} -2 a^3 z^3-2 z^3 a^{-3} -2 a^3 z-2 z a^{-3} - a^{-3} z^{-1} +a z^5+z^5 a^{-1} +a z^3+z^3 a^{-1} +a z+2 z a^{-1} + a^{-1} z^{-1} (db)
Kauffman polynomial -a^2 z^{10}-z^{10}-3 a^3 z^9-6 a z^9-3 z^9 a^{-1} -3 a^4 z^8-4 a^2 z^8-4 z^8 a^{-2} -5 z^8-a^5 z^7+10 a^3 z^7+18 a z^7+3 z^7 a^{-1} -4 z^7 a^{-3} +13 a^4 z^6+26 a^2 z^6+3 z^6 a^{-2} -4 z^6 a^{-4} +20 z^6+4 a^5 z^5-6 a^3 z^5-14 a z^5-z^5 a^{-1} -3 z^5 a^{-5} -16 a^4 z^4-32 a^2 z^4+z^4 a^{-2} +3 z^4 a^{-4} -z^4 a^{-6} -19 z^4-4 a^5 z^3-2 a^3 z^3+4 a z^3+5 z^3 a^{-1} +7 z^3 a^{-3} +4 z^3 a^{-5} +6 a^4 z^2+11 a^2 z^2+z^2 a^{-6} +6 z^2+a^5 z+a^3 z-a z-4 z a^{-1} -5 z a^{-3} -2 z a^{-5} - a^{-2} + a^{-1} z^{-1} + a^{-3} 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 \
-6-5-4-3-2-1012345χ
12           1-1
10          2 2
8         31 -2
6        62  4
4       64   -2
2      75    2
0     67     1
-2    56      -1
-4   36       3
-6  25        -3
-8 13         2
-10 2          -2
-121           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=0 i=2
r=-6 {\mathbb Z}
r=-5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=0 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{7}
r=1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{6}
r=3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
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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L11a195

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L11a197