L10n70

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L10n69.gif

L10n69

L10n71.gif

L10n71

Contents

L10n70.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L10n70's Link Presentations]

Planar diagram presentation X6172 X5,12,6,13 X8493 X2,14,3,13 X14,7,15,8 X9,18,10,19 X11,16,12,17 X17,20,18,11 X4,15,1,16 X19,10,20,5
Gauss code {1, -4, 3, -9}, {-2, -1, 5, -3, -6, 10}, {-7, 2, 4, -5, 9, 7, -8, 6, -10, 8}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L10n70 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^{-9} -2 q^{-8} +2 q^{-7} -2 q^{-6} +3 q^{-5} -2 q^{-4} +3 q^{-3} + q^{-1} (db)
Signature -2 (db)
HOMFLY-PT polynomial z^2 a^8+a^8-z^4 a^6-3 z^2 a^6+a^6 z^{-2} -2 a^6+z^2 a^4-2 a^4 z^{-2} -a^4+z^2 a^2+a^2 z^{-2} +2 a^2 (db)
Kauffman polynomial a^{10} z^6-4 a^{10} z^4+3 a^{10} z^2-a^{10}+2 a^9 z^7-9 a^9 z^5+9 a^9 z^3-2 a^9 z+a^8 z^8-3 a^8 z^6-2 a^8 z^4+6 a^8 z^2-a^8+3 a^7 z^7-14 a^7 z^5+17 a^7 z^3-6 a^7 z+a^6 z^8-4 a^6 z^6+3 a^6 z^4+a^6 z^2+a^6 z^{-2} -a^6+a^5 z^7-5 a^5 z^5+8 a^5 z^3-2 a^5 z-2 a^5 z^{-1} +a^4 z^4-a^4 z^2+2 a^4 z^{-2} -2 a^4+2 a^3 z-2 a^3 z^{-1} +a^2 z^2+a^2 z^{-2} -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 \
-8-7-6-5-4-3-2-10χ
-1        11
-3       121
-5      2 13
-7     12  1
-9    32   1
-11   131   1
-13  121    0
-15 11      0
-17 1       -1
-191        1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-5 i=-3 i=-1
r=-8 {\mathbb Z}
r=-7 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-6 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z} {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=-3 {\mathbb Z} {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}_2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-1 {\mathbb Z}_2 {\mathbb Z}
r=0 {\mathbb Z} {\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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L10n69.gif

L10n69

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L10n71