L10n32

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L10n31.gif

L10n31

L10n33.gif

L10n33

Contents

L10n32.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L10n32's Link Presentations]

Planar diagram presentation X6172 X12,4,13,3 X7,16,8,17 X17,20,18,5 X11,19,12,18 X19,11,20,10 X9,14,10,15 X15,8,16,9 X2536 X4,14,1,13
Gauss code {1, -9, 2, -10}, {9, -1, -3, 8, -7, 6, -5, -2, 10, 7, -8, 3, -4, 5, -6, 4}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gif
A Morse Link Presentation L10n32 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) 0 (db)
Jones polynomial -q^{7/2}+q^{5/2}-q^{3/2}-\frac{1}{\sqrt{q}}+\frac{1}{q^{5/2}}-\frac{1}{q^{7/2}}+\frac{1}{q^{9/2}}-\frac{1}{q^{11/2}} (db)
Signature 0 (db)
HOMFLY-PT polynomial a^5 z+a^5 z^{-1} -a^3 z^3-3 a^3 z-2 a^3 z^{-1} -z a^{-3} - a^{-3} z^{-1} +z^3 a^{-1} +a z^{-1} +3 z a^{-1} + a^{-1} z^{-1} (db)
Kauffman polynomial -a^4 z^8-a^2 z^8-a^5 z^7-2 a^3 z^7-a z^7+6 a^4 z^6+7 a^2 z^6-z^6 a^{-2} +6 a^5 z^5+13 a^3 z^5+7 a z^5-z^5 a^{-1} -z^5 a^{-3} -10 a^4 z^4-14 a^2 z^4+4 z^4 a^{-2} -10 a^5 z^3-23 a^3 z^3-12 a z^3+5 z^3 a^{-1} +4 z^3 a^{-3} +6 a^4 z^2+11 a^2 z^2-2 z^2 a^{-2} +3 z^2+5 a^5 z+13 a^3 z+6 a z-5 z a^{-1} -3 z a^{-3} -a^4-3 a^2- a^{-2} -2-a^5 z^{-1} -2 a^3 z^{-1} -a z^{-1} + a^{-1} z^{-1} + a^{-3} z^{-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 \
-6-5-4-3-2-101234χ
8          11
6           0
4        11 0
2      21   1
0      21   1
-2    122    1
-4   1       -1
-6   11      0
-8 11        0
-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}
r=-3 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z} {\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}_2 {\mathbb Z}^{2}
r=0 {\mathbb Z}^{2} {\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}
r=3 {\mathbb Z}
r=4 {\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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L10n31.gif

L10n31

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L10n33