L11n164

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L11n163.gif

L11n163

L11n165.gif

L11n165

Contents

L11n164.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11n164's Link Presentations]

Planar diagram presentation X8192 X10,3,11,4 X12,17,13,18 X14,5,15,6 X4,13,5,14 X18,11,19,12 X19,7,20,22 X15,21,16,20 X21,17,22,16 X2738 X6,9,1,10
Gauss code {1, -10, 2, -5, 4, -11}, {10, -1, 11, -2, 6, -3, 5, -4, -8, 9, 3, -6, -7, 8, -9, 7}
A Braid Representative
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A Morse Link Presentation L11n164 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{u^2 v^3-u^2 v^2+2 u^2 v-u^2+u v^4-2 u v^3+3 u v^2-2 u v+u-v^4+2 v^3-v^2+v}{u v^2} (db)
Jones polynomial -\frac{6}{q^{9/2}}+\frac{6}{q^{7/2}}-\frac{6}{q^{5/2}}+\frac{4}{q^{3/2}}-\frac{1}{q^{17/2}}+\frac{2}{q^{15/2}}-\frac{4}{q^{13/2}}+\frac{5}{q^{11/2}}+\sqrt{q}-\frac{3}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a^7 z^3+2 a^7 z+a^7 z^{-1} -a^5 z^5-3 a^5 z^3-3 a^5 z-a^5 z^{-1} -a^3 z^5-3 a^3 z^3-3 a^3 z+a z^3+a z (db)
Kauffman polynomial a^9 z^7-5 a^9 z^5+8 a^9 z^3-4 a^9 z+2 a^8 z^8-9 a^8 z^6+11 a^8 z^4-3 a^8 z^2+a^7 z^9-a^7 z^7-8 a^7 z^5+10 a^7 z^3-a^7 z^{-1} +4 a^6 z^8-14 a^6 z^6+11 a^6 z^4-3 a^6 z^2+a^6+a^5 z^9-7 a^5 z^5+3 a^5 z^3+2 a^5 z-a^5 z^{-1} +2 a^4 z^8-4 a^4 z^6+2 a^3 z^7-4 a^3 z^5+4 a^3 z^3-3 a^3 z+a^2 z^6+a^2 z^2+3 a z^3-a z+z^2 (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 \
-8-7-6-5-4-3-2-101χ
2         1-1
0        2 2
-2       32 -1
-4      31  2
-6     33   0
-8    33    0
-10   23     1
-12  23      -1
-14 13       2
-16 1        -1
-181         1
Integral Khovanov Homology

(db, data source)

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