L10n19

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L10n18

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L10n20

Contents

L10n19.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L10n19's Link Presentations]

Planar diagram presentation X6172 X18,7,19,8 X4,19,1,20 X11,14,12,15 X3849 X5,13,6,12 X13,5,14,20 X9,16,10,17 X15,10,16,11 X17,2,18,3
Gauss code {1, 10, -5, -3}, {-6, -1, 2, 5, -8, 9, -4, 6, -7, 4, -9, 8, -10, -2, 3, 7}
A Braid Representative
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A Morse Link Presentation L10n19 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{2 t(1) t(2)^3-4 t(1) t(2)^2+t(2)^2+t(1) t(2)-4 t(2)+2}{\sqrt{t(1)} t(2)^{3/2}} (db)
Jones polynomial \frac{5}{q^{9/2}}-\frac{5}{q^{7/2}}+\frac{4}{q^{5/2}}-\frac{3}{q^{3/2}}-\frac{2}{q^{15/2}}+\frac{3}{q^{13/2}}-\frac{4}{q^{11/2}}-\sqrt{q}+\frac{1}{\sqrt{q}} (db)
Signature -3 (db)
HOMFLY-PT polynomial a^9 z^{-1} -a^7 z^3-3 a^7 z-2 a^7 z^{-1} +a^5 z^5+3 a^5 z^3+2 a^5 z+a^5 z^{-1} +a^3 z^5+3 a^3 z^3+2 a^3 z+a^3 z^{-1} -a z^3-3 a z-a z^{-1} (db)
Kauffman polynomial 3 a^9 z^3-5 a^9 z+a^9 z^{-1} +a^8 z^6-2 a^8 z^2+a^8+2 a^7 z^7-8 a^7 z^5+18 a^7 z^3-13 a^7 z+2 a^7 z^{-1} +a^6 z^8-3 a^6 z^6+8 a^6 z^4-7 a^6 z^2+3 a^6+3 a^5 z^7-10 a^5 z^5+16 a^5 z^3-9 a^5 z+a^5 z^{-1} +a^4 z^8-3 a^4 z^6+6 a^4 z^4-6 a^4 z^2+2 a^4+a^3 z^7-a^3 z^5-3 a^3 z^3+3 a^3 z-a^3 z^{-1} +a^2 z^6-2 a^2 z^4-a^2 z^2+a^2+a z^5-4 a z^3+4 a z-a 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 \
-6-5-4-3-2-1012χ
2        11
0         0
-2      31 2
-4     21  -1
-6    32   1
-8   22    0
-10  23     -1
-12 23      1
-14 1       -1
-162        2
Integral Khovanov Homology

(db, data source)

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

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L10n18

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