L11n8

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L11n7.gif

L11n7

L11n9.gif

L11n9

Contents

L11n8.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n8 at Knotilus!


Link Presentations

[edit Notes on L11n8's Link Presentations]

Planar diagram presentation X6172 X16,7,17,8 X17,1,18,4 X9,14,10,15 X3849 X5,11,6,10 X11,20,12,21 X19,22,20,5 X13,19,14,18 X21,12,22,13 X2,16,3,15
Gauss code {1, -11, -5, 3}, {-6, -1, 2, 5, -4, 6, -7, 10, -9, 4, 11, -2, -3, 9, -8, 7, -10, 8}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart3.gif
BraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11n8 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(u-1) (v-1)^3}{\sqrt{u} v^{3/2}} (db)
Jones polynomial -q^{3/2}+2 \sqrt{q}-\frac{4}{\sqrt{q}}+\frac{4}{q^{3/2}}-\frac{6}{q^{5/2}}+\frac{5}{q^{7/2}}-\frac{4}{q^{9/2}}+\frac{3}{q^{11/2}}-\frac{2}{q^{13/2}}+\frac{1}{q^{15/2}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a^7 (-z)-a^7 z^{-1} +2 a^5 z^3+5 a^5 z+3 a^5 z^{-1} -a^3 z^5-4 a^3 z^3-7 a^3 z-3 a^3 z^{-1} +2 a z^3+4 a z+2 a z^{-1} -z a^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial -a^5 z^9-a^3 z^9-2 a^6 z^8-4 a^4 z^8-2 a^2 z^8-2 a^7 z^7+a^5 z^7+2 a^3 z^7-a z^7-a^8 z^6+7 a^6 z^6+18 a^4 z^6+10 a^2 z^6+8 a^7 z^5+10 a^5 z^5+6 a^3 z^5+4 a z^5+4 a^8 z^4-3 a^6 z^4-24 a^4 z^4-19 a^2 z^4-2 z^4-7 a^7 z^3-19 a^5 z^3-21 a^3 z^3-10 a z^3-z^3 a^{-1} -3 a^8 z^2-3 a^6 z^2+8 a^4 z^2+11 a^2 z^2+3 z^2+3 a^7 z+12 a^5 z+16 a^3 z+9 a z+2 z a^{-1} +a^8+2 a^6-2 a^2-a^7 z^{-1} -3 a^5 z^{-1} -3 a^3 z^{-1} -2 a z^{-1} - a^{-1} 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 \
-7-6-5-4-3-2-1012χ
4         11
2        1 -1
0       31 2
-2      33  0
-4     321  2
-6    34    1
-8   23     -1
-10  131     1
-12 12       -1
-14 1        1
-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}\oplus{\mathbb Z}_2 {\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} {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-2 {\mathbb Z}_2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=0 {\mathbb Z} {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
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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