L10n98

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L10n97.gif

L10n97

L10n99.gif

L10n99

Contents

L10n98.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L10n98's Link Presentations]

Planar diagram presentation X6172 X2536 X13,15,14,20 X10,3,11,4 X4,9,1,10 X16,7,17,8 X8,15,5,16 X11,19,12,18 X19,13,20,12 X17,9,18,14
Gauss code {1, -2, 4, -5}, {2, -1, 6, -7}, {5, -4, -8, 9, -3, 10}, {7, -6, -10, 8, -9, 3}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart3.gif
BraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart1.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart2.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L10n98 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(1) t(4)^2 t(3)^2+t(2) t(4)^2 t(3)^2-t(4)^2 t(3)^2-t(2) t(4) t(3)^2-t(1) t(4)^2 t(3)-t(2) t(3)+t(1) t(4) t(3)+t(2) t(4) t(3)+t(1)-t(1) t(2)+t(2)-t(1) t(4)}{\sqrt{t(1)} \sqrt{t(2)} t(3) t(4)} (db)
Jones polynomial \frac{1}{q^{9/2}}-\frac{4}{q^{7/2}}-2 q^{5/2}+\frac{2}{q^{5/2}}+2 q^{3/2}-\frac{5}{q^{3/2}}-\frac{1}{q^{11/2}}-4 \sqrt{q}+\frac{3}{\sqrt{q}} (db)
Signature 1 (db)
HOMFLY-PT polynomial a^5 z^{-3} +a^5 z+3 a^5 z^{-1} -3 a^3 z^3-3 a^3 z^{-3} -11 a^3 z-10 a^3 z^{-1} +2 a z^5+10 a z^3+3 a z^{-3} -2 z^3 a^{-1} - a^{-1} z^{-3} +16 a z+11 a z^{-1} -6 z a^{-1} -4 a^{-1} z^{-1} (db)
Kauffman polynomial -a^4 z^8-a^2 z^8-a^5 z^7-5 a^3 z^7-4 a z^7+3 a^4 z^6-a^2 z^6-4 z^6+6 a^5 z^5+25 a^3 z^5+17 a z^5-2 z^5 a^{-1} +4 a^4 z^4+19 a^2 z^4-z^4 a^{-2} +14 z^4-13 a^5 z^3-39 a^3 z^3-24 a z^3+2 z^3 a^{-1} -17 a^4 z^2-33 a^2 z^2-16 z^2+13 a^5 z+28 a^3 z+21 a z+3 z a^{-1} -3 z a^{-3} +13 a^4+24 a^2- a^{-2} +11-6 a^5 z^{-1} -14 a^3 z^{-1} -12 a z^{-1} -3 a^{-1} z^{-1} + a^{-3} z^{-1} -3 a^4 z^{-2} -6 a^2 z^{-2} -3 z^{-2} +a^5 z^{-3} +3 a^3 z^{-3} +3 a z^{-3} + a^{-1} z^{-3} (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
6        22
4       330
2      1 12
0     34  1
-2    211  2
-4   14    3
-6  31     2
-8 14      3
-10         0
-121        1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-2 i=0 i=2
r=-6 {\mathbb Z}
r=-5 {\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{4} {\mathbb Z}^{3}
r=-3 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=-1 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=0 {\mathbb Z} {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}
r=1 {\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=2 {\mathbb Z} {\mathbb Z}^{3} {\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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L10n97

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L10n99