L11n197

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L11n196.gif

L11n196

L11n198.gif

L11n198

Contents

L11n197.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n197 at Knotilus!


Link Presentations

[edit Notes on L11n197's Link Presentations]

Planar diagram presentation X8192 X12,3,13,4 X22,10,7,9 X10,14,11,13 X5,19,6,18 X21,16,22,17 X15,20,16,21 X19,14,20,15 X2738 X4,11,5,12 X17,1,18,6
Gauss code {1, -9, 2, -10, -5, 11}, {9, -1, 3, -4, 10, -2, 4, 8, -7, 6, -11, 5, -8, 7, -6, -3}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gif
BraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart2.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11n197 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{u^2 v^2-u^2+u v^4-u v^3-u v^2-u v+u-v^4+v^2}{u v^2} (db)
Jones polynomial -\frac{1}{q^{9/2}}-\frac{1}{q^{7/2}}-q^{5/2}+2 q^{3/2}-\frac{2}{q^{3/2}}+\frac{1}{q^{15/2}}-\frac{1}{q^{13/2}}+\frac{1}{q^{11/2}}-2 \sqrt{q}+\frac{2}{\sqrt{q}} (db)
Signature -3 (db)
HOMFLY-PT polynomial a^7 (-z)-a^7 z^{-1} +a^5 z^3+3 a^5 z+3 a^5 z^{-1} -a^3 z^3-4 a^3 z-2 a^3 z^{-1} +a z^5+4 a z^3-z^3 a^{-1} +3 a z-2 z a^{-1} (db)
Kauffman polynomial -a^3 z^9-a z^9-a^4 z^8-3 a^2 z^8-2 z^8-a^7 z^7+6 a^3 z^7+4 a z^7-z^7 a^{-1} -a^8 z^6-a^6 z^6+7 a^4 z^6+18 a^2 z^6+11 z^6+5 a^7 z^5+a^5 z^5-9 a^3 z^5+5 z^5 a^{-1} +5 a^8 z^4+6 a^6 z^4-11 a^4 z^4-28 a^2 z^4-16 z^4-6 a^7 z^3-a^5 z^3+6 a^3 z^3-5 a z^3-6 z^3 a^{-1} -6 a^8 z^2-7 a^6 z^2+4 a^4 z^2+12 a^2 z^2+7 z^2+3 a^7 z+4 a^5 z-a^3 z+2 z a^{-1} +a^8+3 a^6+3 a^4-a^7 z^{-1} -3 a^5 z^{-1} -2 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 \
-7-6-5-4-3-2-101234χ
6           11
4          1 -1
2         11 0
0       121  0
-2      121   0
-4     112    2
-6    131     1
-8   1 1      2
-10   11       0
-12 11         0
-14            0
-161           -1
Integral Khovanov Homology

(db, data source)

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

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See/edit the Link Page master template (intermediate).

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L11n196

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L11n198