L10a83

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L10a82.gif

L10a82

L10a84.gif

L10a84

Contents

L10a83.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L10a83's Link Presentations]

Planar diagram presentation X8192 X12,3,13,4 X20,10,7,9 X14,17,15,18 X16,5,17,6 X4,15,5,16 X18,13,19,14 X10,20,11,19 X2738 X6,11,1,12
Gauss code {1, -9, 2, -6, 5, -10}, {9, -1, 3, -8, 10, -2, 7, -4, 6, -5, 4, -7, 8, -3}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
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BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gif
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A Morse Link Presentation L10a83 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{2 t(2)^2 t(1)^2-5 t(2) t(1)^2+2 t(1)^2-5 t(2)^2 t(1)+9 t(2) t(1)-5 t(1)+2 t(2)^2-5 t(2)+2}{t(1) t(2)} (db)
Jones polynomial -\sqrt{q}+\frac{3}{\sqrt{q}}-\frac{6}{q^{3/2}}+\frac{9}{q^{5/2}}-\frac{12}{q^{7/2}}+\frac{12}{q^{9/2}}-\frac{12}{q^{11/2}}+\frac{9}{q^{13/2}}-\frac{6}{q^{15/2}}+\frac{3}{q^{17/2}}-\frac{1}{q^{19/2}} (db)
Signature -3 (db)
HOMFLY-PT polynomial z a^9-2 z^3 a^7-z a^7+a^7 z^{-1} +z^5 a^5-2 z a^5-a^5 z^{-1} +z^5 a^3+z^3 a^3-z^3 a-z a (db)
Kauffman polynomial a^{11} z^5-2 a^{11} z^3+a^{11} z+3 a^{10} z^6-6 a^{10} z^4+3 a^{10} z^2+4 a^9 z^7-6 a^9 z^5+2 a^9 z^3-a^9 z+3 a^8 z^8-6 a^8 z^4+3 a^8 z^2+a^7 z^9+7 a^7 z^7-14 a^7 z^5+7 a^7 z^3+a^7 z-a^7 z^{-1} +6 a^6 z^8-6 a^6 z^6+a^6+a^5 z^9+7 a^5 z^7-14 a^5 z^5+7 a^5 z^3+a^5 z-a^5 z^{-1} +3 a^4 z^8-6 a^4 z^4+3 a^4 z^2+4 a^3 z^7-6 a^3 z^5+2 a^3 z^3-a^3 z+3 a^2 z^6-6 a^2 z^4+3 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 \
-8-7-6-5-4-3-2-1012χ
2          11
0         2 -2
-2        41 3
-4       63  -3
-6      63   3
-8     66    0
-10    66     0
-12   36      3
-14  36       -3
-16 14        3
-18 2         -2
-201          1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-4 i=-2
r=-8 {\mathbb Z}
r=-7 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-6 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=-5 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-4 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=0 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{4}
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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L10a82

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L10a84