L11n88

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L11n87.gif

L11n87

L11n89.gif

L11n89

Contents

L11n88.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n88 at Knotilus!


Link Presentations

[edit Notes on L11n88's Link Presentations]

Planar diagram presentation X6172 X3,13,4,12 X7,16,8,17 X17,22,18,5 X9,15,10,14 X19,10,20,11 X21,9,22,8 X13,18,14,19 X15,21,16,20 X2536 X11,1,12,4
Gauss code {1, -10, -2, 11}, {10, -1, -3, 7, -5, 6, -11, 2, -8, 5, -9, 3, -4, 8, -6, 9, -7, 4}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gif
BraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11n88 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(u-1) (v-1)^3 \left(v^2-v+1\right)}{\sqrt{u} v^{5/2}} (db)
Jones polynomial q^{9/2}-\frac{3}{q^{9/2}}-4 q^{7/2}+\frac{8}{q^{7/2}}+8 q^{5/2}-\frac{12}{q^{5/2}}-13 q^{3/2}+\frac{15}{q^{3/2}}+15 \sqrt{q}-\frac{17}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a z^7-a^3 z^5+4 a z^5-2 z^5 a^{-1} -3 a^3 z^3+7 a z^3-5 z^3 a^{-1} +z^3 a^{-3} +a^5 z-5 a^3 z+7 a z-4 z a^{-1} +z a^{-3} +a^5 z^{-1} -3 a^3 z^{-1} +4 a z^{-1} -2 a^{-1} z^{-1} (db)
Kauffman polynomial 6 a^5 z^3-4 a^5 z+a^5 z^{-1} +3 a^4 z^6+z^6 a^{-4} +6 a^4 z^4-2 z^4 a^{-4} -7 a^4 z^2+z^2 a^{-4} +a^4+10 a^3 z^7+4 z^7 a^{-3} -21 a^3 z^5-10 z^5 a^{-3} +28 a^3 z^3+8 z^3 a^{-3} -16 a^3 z-3 z a^{-3} +3 a^3 z^{-1} +10 a^2 z^8+6 z^8 a^{-2} -22 a^2 z^6-13 z^6 a^{-2} +27 a^2 z^4+5 z^4 a^{-2} -17 a^2 z^2+3 a^2+ a^{-2} +3 a z^9+3 z^9 a^{-1} +13 a z^7+7 z^7 a^{-1} -48 a z^5-37 z^5 a^{-1} +49 a z^3+35 z^3 a^{-1} -22 a z-13 z a^{-1} +4 a z^{-1} +2 a^{-1} z^{-1} +16 z^8-39 z^6+28 z^4-11 z^2+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 \
-4-3-2-1012345χ
10         1-1
8        3 3
6       51 -4
4      83  5
2     75   -2
0    108    2
-2   79     2
-4  58      -3
-6 37       4
-8 5        -5
-103         3
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-2 i=0
r=-4 {\mathbb Z}^{3}
r=-3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=0 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{10}
r=1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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

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

See/edit the Link_Splice_Base (expert).

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L11n87

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L11n89