L8n8

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L8n7.gif

L8n7

L9a1.gif

L9a1

Contents

L8n8.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L8n8 at Knotilus!

L8n8 is 8^4_{3} in the Rolfsen table of links.

Detail from an 18th century royal decree, Vietnam.

Link Presentations

[edit Notes on L8n8's Link Presentations]

Planar diagram presentation X6172 X2536 X16,11,13,12 X3,11,4,10 X9,1,10,4 X7,15,8,14 X13,5,14,8 X12,15,9,16
Gauss code {1, -2, -4, 5}, {2, -1, -6, 7}, {-5, 4, 3, -8}, {-7, 6, 8, -3}
A Braid Representative
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A Morse Link Presentation L8n8 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(t(2)-t(3)) (t(1)-t(4))}{\sqrt{t(1)} \sqrt{t(2)} \sqrt{t(3)} \sqrt{t(4)}} (db)
Jones polynomial -q^{7/2}-q^{3/2}-2 \sqrt{q}-\frac{2}{\sqrt{q}}-\frac{1}{q^{3/2}}-\frac{1}{q^{7/2}} (db)
Signature 0 (db)
HOMFLY-PT polynomial a^3 z^{-3} - a^{-3} z^{-3} +a^3 z+2 a^3 z^{-1} -z a^{-3} -2 a^{-3} z^{-1} -a z^3-3 a z^{-3} +z^3 a^{-1} +3 a^{-1} z^{-3} -5 a z-6 a z^{-1} +5 z a^{-1} +6 a^{-1} z^{-1} (db)
Kauffman polynomial -a^3 z^5-a z^5-z^5 a^{-1} -z^5 a^{-3} -a^2 z^4-z^4 a^{-2} -2 z^4+5 a^3 z^3+7 a z^3+7 z^3 a^{-1} +5 z^3 a^{-3} +6 a^2 z^2+6 z^2 a^{-2} +12 z^2-6 a^3 z-14 a z-14 z a^{-1} -6 z a^{-3} -8 a^2-8 a^{-2} -15+4 a^3 z^{-1} +9 a z^{-1} +9 a^{-1} z^{-1} +4 a^{-3} z^{-1} +3 a^2 z^{-2} +3 a^{-2} z^{-2} +6 z^{-2} -a^3 z^{-3} -3 a z^{-3} -3 a^{-1} z^{-3} - a^{-3} z^{-3} (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 \
-4-3-2-101234χ
8        11
6        11
4      1  1
2    3    3
0   161   4
-2    3    3
-4  1      1
-61        1
-81        1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-2 i=0 i=2
r=-4 {\mathbb Z} {\mathbb Z}
r=-3
r=-2 {\mathbb Z}
r=-1 {\mathbb Z}_2 {\mathbb Z}
r=0 {\mathbb Z}^{3} {\mathbb Z}^{6} {\mathbb Z}^{3}
r=1 {\mathbb Z}
r=2 {\mathbb Z}_2 {\mathbb Z}
r=3
r=4 {\mathbb Z} {\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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L8n7.gif

L8n7

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L9a1