L8n5

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L8n4.gif

L8n4

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L8n6

Contents

L8n5.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L8n5 at Knotilus!

L8n5 is is 8^3_{9} in the Rolfsen table of links.


Link Presentations

[edit Notes on L8n5's Link Presentations]

Planar diagram presentation X6172 X5,12,6,13 X8493 X2,14,3,13 X14,7,15,8 X9,16,10,11 X11,10,12,5 X4,15,1,16
Gauss code {1, -4, 3, -8}, {-2, -1, 5, -3, -6, 7}, {-7, 2, 4, -5, 8, 6}
A Braid Representative
BraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart3.gifBraidPart3.gif
BraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart4.gif
A Morse Link Presentation L8n5 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(t(1)-1) (t(2)-1) (t(3)-1)}{\sqrt{t(1)} \sqrt{t(2)} \sqrt{t(3)}} (db)
Jones polynomial  q^{-7} -2 q^{-6} +3 q^{-5} -2 q^{-4} +4 q^{-3} -2 q^{-2} +2 q^{-1} (db)
Signature -2 (db)
HOMFLY-PT polynomial a^6 z^2+a^6 z^{-2} +a^6-a^4 z^4-3 a^4 z^2-2 a^4 z^{-2} -4 a^4+2 a^2 z^2+a^2 z^{-2} +3 a^2 (db)
Kauffman polynomial a^8 z^4-2 a^8 z^2+a^8+2 a^7 z^5-4 a^7 z^3+a^6 z^6-2 a^6 z^2+a^6 z^{-2} -a^6+3 a^5 z^5-5 a^5 z^3+4 a^5 z-2 a^5 z^{-1} +a^4 z^6-a^4 z^4+3 a^4 z^2+2 a^4 z^{-2} -4 a^4+a^3 z^5-a^3 z^3+4 a^3 z-2 a^3 z^{-1} +3 a^2 z^2+a^2 z^{-2} -3 a^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 \
-6-5-4-3-2-10χ
-1      22
-3     220
-5    2  2
-7    2  2
-9  32   1
-11 12    1
-13 1     -1
-151      1
Integral Khovanov Homology

(db, data source)

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

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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