L10n82

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L10n81.gif

L10n81

L10n83.gif

L10n83

Contents

L10n82.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L10n82 at Knotilus!


Link Presentations

[edit Notes on L10n82's Link Presentations]

Planar diagram presentation X6172 X14,4,15,3 X11,20,12,13 X7,18,8,19 X9,16,10,17 X17,8,18,9 X13,10,14,11 X19,12,20,5 X2536 X4,16,1,15
Gauss code {1, -9, 2, -10}, {9, -1, -4, 6, -5, 7, -3, 8}, {-7, -2, 10, 5, -6, 4, -8, 3}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gif
A Morse Link Presentation L10n82 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{u v^2 w^2-u v^2 w-u v w^2+u v w+u w^3-v^3-v^2 w^2+v^2 w+v w^2-v w}{\sqrt{u} v^{3/2} w^{3/2}} (db)
Jones polynomial  q^{-8} + q^{-6} + q^{-5} -2 q^{-4} +3 q^{-3} -2 q^{-2} +q+3 q^{-1} -2 (db)
Signature -2 (db)
HOMFLY-PT polynomial a^8 z^{-2} +a^8-a^6 z^2-2 a^6 z^{-2} -3 a^6+a^4 z^2+a^4 z^{-2} +a^4-a^2 z^4-2 a^2 z^2+z^2+1 (db)
Kauffman polynomial a^8 z^8-8 a^8 z^6+21 a^8 z^4-22 a^8 z^2-a^8 z^{-2} +8 a^8+a^7 z^7-8 a^7 z^5+17 a^7 z^3-11 a^7 z+2 a^7 z^{-1} +a^6 z^8-9 a^6 z^6+25 a^6 z^4-27 a^6 z^2-2 a^6 z^{-2} +13 a^6+a^5 z^7-8 a^5 z^5+19 a^5 z^3-11 a^5 z+2 a^5 z^{-1} +3 a^4 z^4-6 a^4 z^2-a^4 z^{-2} +5 a^4+2 a^3 z^5-2 a^3 z^3+a^2 z^6-3 a^2 z^2+2 a z^5-4 a z^3+z^4-2 z^2+1 (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 \
-8-7-6-5-4-3-2-1012χ
3          11
1         1 -1
-1        21 1
-3       23  1
-5     12    1
-7     12    1
-9   122     -1
-11    2      2
-13  1        1
-151          1
-171          1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-3 i=-1 i=1
r=-8 {\mathbb Z} {\mathbb Z}
r=-7
r=-6 {\mathbb Z}
r=-5 {\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{2} {\mathbb Z}^{2}
r=-3 {\mathbb Z}^{2} {\mathbb Z} {\mathbb Z}
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}^{3} {\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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L10n83