L10n103

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L10n102.gif

L10n102

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L10n104

Contents

L10n103.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L10n103's Link Presentations]

Planar diagram presentation X6172 X10,3,11,4 X14,7,15,8 X8,13,5,14 X11,18,12,19 X20,16,17,15 X16,20,9,19 X17,12,18,13 X2536 X4,9,1,10
Gauss code {1, -9, 2, -10}, {9, -1, 3, -4}, {-8, 5, 7, -6}, {10, -2, -5, 8, 4, -3, 6, -7}
A Braid Representative
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A Morse Link Presentation L10n103 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(t(1)-1) (t(2)-1) (t(3)-1) (t(4)-1)}{\sqrt{t(1)} \sqrt{t(2)} \sqrt{t(3)} \sqrt{t(4)}} (db)
Jones polynomial -\sqrt{q}+\frac{3}{\sqrt{q}}-\frac{5}{q^{3/2}}+\frac{4}{q^{5/2}}-\frac{7}{q^{7/2}}+\frac{4}{q^{9/2}}-\frac{6}{q^{11/2}}+\frac{1}{q^{13/2}}-\frac{1}{q^{15/2}} (db)
Signature -3 (db)
HOMFLY-PT polynomial a^9 z^{-3} -3 a^7 z^{-3} -3 a^7 z^{-1} -a^5 z^3+3 a^5 z^{-3} +2 a^5 z+6 a^5 z^{-1} +a^3 z^5+2 a^3 z^3-a^3 z^{-3} -a^3 z-3 a^3 z^{-1} -a z^3-a z (db)
Kauffman polynomial a^9 z^3-a^9 z^{-3} -3 a^9 z+3 a^9 z^{-1} +a^8 z^4+2 a^8 z^2+3 a^8 z^{-2} -6 a^8+a^7 z^7-4 a^7 z^5+12 a^7 z^3-3 a^7 z^{-3} -9 a^7 z+6 a^7 z^{-1} +a^6 z^8-2 a^6 z^6+a^6 z^4+9 a^6 z^2+6 a^6 z^{-2} -11 a^6+4 a^5 z^7-12 a^5 z^5+16 a^5 z^3-3 a^5 z^{-3} -11 a^5 z+6 a^5 z^{-1} +a^4 z^8+a^4 z^6-7 a^4 z^4+8 a^4 z^2+3 a^4 z^{-2} -6 a^4+3 a^3 z^7-7 a^3 z^5+3 a^3 z^3-a^3 z^{-3} -4 a^3 z+3 a^3 z^{-1} +3 a^2 z^6-7 a^2 z^4+a^2 z^2+a z^5-2 a z^3+a z (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       2 -2
-2      31 2
-4     34  1
-6    41   3
-8  113    3
-10  64     2
-12 16      5
-14         0
-161        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}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{6} {\mathbb Z}^{6} {\mathbb Z}
r=-3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-1 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=0 {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}^{3}
r=1 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
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

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L10n102.gif

L10n102

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L10n104