L8n4

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L8n3.gif

L8n3

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L8n5

Contents

L8n4.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L8n4 at Knotilus!

L8n4 is 8^3_{8} in the Rolfsen table of links.


Link Presentations

[edit Notes on L8n4's Link Presentations]

Planar diagram presentation X6172 X5,12,6,13 X3849 X2,14,3,13 X14,7,15,8 X9,16,10,11 X11,10,12,5 X15,1,16,4
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.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart3.gif
BraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart4.gif
A Morse Link Presentation L8n4 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(3) t(2)^2-t(1) t(3)^2 t(2)+t(1) t(3) t(2)-t(3) t(2)+t(2)-t(1) t(3)}{\sqrt{t(1)} t(2) t(3)} (db)
Jones polynomial  q^{-7} - q^{-6} +2 q^{-5} -2 q^{-4} +3 q^{-3} - q^{-2} +2 q^{-1} (db)
Signature -2 (db)
HOMFLY-PT polynomial a^6 z^2+a^6 z^{-2} +2 a^6-a^4 z^4-4 a^4 z^2-2 a^4 z^{-2} -6 a^4+2 a^2 z^2+a^2 z^{-2} +4 a^2 (db)
Kauffman polynomial z^4 a^8-3 z^2 a^8+a^8+z^5 a^7-2 z^3 a^7+z^6 a^6-3 z^4 a^6+4 z^2 a^6+a^6 z^{-2} -3 a^6+2 z^5 a^5-5 z^3 a^5+6 z a^5-2 a^5 z^{-1} +z^6 a^4-4 z^4 a^4+10 z^2 a^4+2 a^4 z^{-2} -8 a^4+z^5 a^3-3 z^3 a^3+6 z a^3-2 a^3 z^{-1} +3 z^2 a^2+a^2 z^{-2} -5 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     121
-5    2  2
-7   12  1
-9  11   0
-11  1    1
-1311     0
-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} {\mathbb Z}
r=-5 {\mathbb Z}
r=-4 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=-1 {\mathbb Z}_2 {\mathbb Z}
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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L8n3

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L8n5