L10n14

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L10n13.gif

L10n13

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L10n15

Contents

L10n14.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L10n14's Link Presentations]

Planar diagram presentation X6172 X18,7,19,8 X4,19,1,20 X9,14,10,15 X8493 X5,13,6,12 X13,5,14,20 X11,16,12,17 X15,10,16,11 X2,18,3,17
Gauss code {1, -10, 5, -3}, {-6, -1, 2, -5, -4, 9, -8, 6, -7, 4, -9, 8, 10, -2, 3, 7}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gif
BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gif
A Morse Link Presentation L10n14 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(t(1)-1) (t(2)-1)}{\sqrt{t(1)} \sqrt{t(2)}} (db)
Jones polynomial -q^{5/2}+q^{3/2}-\sqrt{q}-\frac{1}{\sqrt{q}}+\frac{1}{q^{3/2}}-\frac{2}{q^{5/2}}+\frac{2}{q^{7/2}}-\frac{2}{q^{9/2}}+\frac{2}{q^{11/2}}-\frac{1}{q^{13/2}} (db)
Signature -1 (db)
HOMFLY-PT polynomial z^3 a^5+2 z a^5+a^5 z^{-1} -z^5 a^3-5 z^3 a^3-8 z a^3-3 a^3 z^{-1} +z^5 a+6 z^3 a+9 z a+4 a z^{-1} -z^3 a^{-1} -3 z a^{-1} -2 a^{-1} z^{-1} (db)
Kauffman polynomial a^7 z^5-3 a^7 z^3+a^7 z+2 a^6 z^6-7 a^6 z^4+4 a^6 z^2-a^6+a^5 z^7-3 a^5 z^5+a^5 z^3-2 a^5 z+a^5 z^{-1} +2 a^4 z^6-9 a^4 z^4+10 a^4 z^2-3 a^4+2 a^3 z^7-13 a^3 z^5+24 a^3 z^3-15 a^3 z+3 a^3 z^{-1} +a^2 z^8-6 a^2 z^6+6 a^2 z^4+5 a^2 z^2-3 a^2+2 a z^7+z^7 a^{-1} -15 a z^5-6 z^5 a^{-1} +30 a z^3+10 z^3 a^{-1} -19 a z-7 z a^{-1} +4 a z^{-1} +2 a^{-1} z^{-1} +z^8-6 z^6+8 z^4-z^2-2 (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 \
-6-5-4-3-2-101234χ
6          11
4           0
2        11 0
0      31   2
-2     121   0
-4    221    1
-6   11      0
-8  121      0
-10 11        0
-12 1         -1
-141          1
Integral Khovanov Homology

(db, data source)

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