L10n83

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L10n82.gif

L10n82

L10n84.gif

L10n84

Contents

L10n83.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L10n83 at Knotilus!


Link Presentations

[edit Notes on L10n83's Link Presentations]

Planar diagram presentation X6172 X12,7,13,8 X4,13,1,14 X5,16,6,17 X8493 X17,20,18,15 X19,10,20,11 X9,18,10,19 X15,14,16,5 X2,12,3,11
Gauss code {1, -10, 5, -3}, {-9, 4, -6, 8, -7, 6}, {-4, -1, 2, -5, -8, 7, 10, -2, 3, 9}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart3.gif
BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart3.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart4.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L10n83 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(u-1) (w-1)^2 (v w+1)}{\sqrt{u} \sqrt{v} w^{3/2}} (db)
Jones polynomial 1-2 q^{-1} +5 q^{-2} -4 q^{-3} +6 q^{-4} -5 q^{-5} +5 q^{-6} -3 q^{-7} + q^{-8} (db)
Signature -4 (db)
HOMFLY-PT polynomial z^4 a^6+2 z^2 a^6+a^6 z^{-2} +a^6-z^6 a^4-4 z^4 a^4-5 z^2 a^4-2 a^4 z^{-2} -4 a^4+z^4 a^2+3 z^2 a^2+a^2 z^{-2} +3 a^2 (db)
Kauffman polynomial z^2 a^{10}-a^{10}+3 z^3 a^9-z a^9+2 z^6 a^8-5 z^4 a^8+9 z^2 a^8-3 a^8+3 z^7 a^7-9 z^5 a^7+10 z^3 a^7-3 z a^7+z^8 a^6+3 z^6 a^6-18 z^4 a^6+20 z^2 a^6+a^6 z^{-2} -7 a^6+5 z^7 a^5-15 z^5 a^5+9 z^3 a^5+z a^5-2 a^5 z^{-1} +z^8 a^4+2 z^6 a^4-17 z^4 a^4+18 z^2 a^4+2 a^4 z^{-2} -8 a^4+2 z^7 a^3-6 z^5 a^3+2 z^3 a^3+3 z a^3-2 a^3 z^{-1} +z^6 a^2-4 z^4 a^2+6 z^2 a^2+a^2 z^{-2} -4 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      41 3
-5     23  1
-7    42   2
-9  122    1
-11  44     0
-13 13      2
-15 2       -2
-171        1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-5 i=-3 i=-1
r=-6 {\mathbb Z}
r=-5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{4} {\mathbb Z}
r=-3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=0 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{4}
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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L10n82

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L10n84