L10n86

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L10n85

L10n87.gif

L10n87

Contents

L10n86.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link L10n86.
A graph, L10n86.
A part of a link and a part of a graph.

Link Presentations

[edit Notes on L10n86's Link Presentations]

Planar diagram presentation X6172 X10,3,11,4 X13,20,14,15 X16,8,17,7 X8,16,9,15 X11,18,12,19 X19,12,20,13 X17,14,18,5 X2536 X4,9,1,10
Gauss code {1, -9, 2, -10}, {5, -4, -8, 6, -7, 3}, {9, -1, 4, -5, 10, -2, -6, 7, -3, 8}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gif
BraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L10n86 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(t(3)-1) \left(t(1) t(3) t(2)^2-t(3) t(2)^2-t(1) t(3) t(2)-t(2)-t(1)+1\right)}{\sqrt{t(1)} t(2) t(3)} (db)
Jones polynomial 1-2 q^{-1} +3 q^{-2} -3 q^{-3} +5 q^{-4} -3 q^{-5} +4 q^{-6} -2 q^{-7} + q^{-8} (db)
Signature -4 (db)
HOMFLY-PT polynomial a^8 z^{-2} +z^4 a^6+2 z^2 a^6-2 a^6 z^{-2} -2 a^6-z^6 a^4-4 z^4 a^4-3 z^2 a^4+a^4 z^{-2} +a^4+z^4 a^2+3 z^2 a^2+a^2 (db)
Kauffman polynomial z^2 a^{10}+2 z^3 a^9+4 z^4 a^8-6 z^2 a^8-a^8 z^{-2} +4 a^8+z^7 a^7-2 z^5 a^7+4 z^3 a^7-5 z a^7+2 a^7 z^{-1} +z^8 a^6-3 z^6 a^6+6 z^4 a^6-10 z^2 a^6-2 a^6 z^{-2} +6 a^6+3 z^7 a^5-10 z^5 a^5+9 z^3 a^5-5 z a^5+2 a^5 z^{-1} +z^8 a^4-2 z^6 a^4-2 z^4 a^4+z^2 a^4-a^4 z^{-2} +2 a^4+2 z^7 a^3-8 z^5 a^3+7 z^3 a^3+z^6 a^2-4 z^4 a^2+4 z^2 a^2-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 \
-6-5-4-3-2-1012χ
1        11
-1       1 -1
-3      21 1
-5     22  0
-7    31   2
-9   13    2
-11  32     1
-13  2      2
-1512       -1
-171        1
Integral Khovanov Homology

(db, data source)

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