L10n88

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L10n87.gif

L10n87

L10n89.gif

L10n89

Contents

L10n88.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L10n88 at Knotilus!


Link Presentations

[edit Notes on L10n88's Link Presentations]

Planar diagram presentation X6172 X10,3,11,4 X7,14,8,15 X15,17,16,20 X11,19,12,18 X17,13,18,12 X19,5,20,16 X13,8,14,9 X2536 X4,9,1,10
Gauss code {1, -9, 2, -10}, {-6, 5, -7, 4}, {9, -1, -3, 8, 10, -2, -5, 6, -8, 3, -4, 7}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
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BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart1.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L10n88 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(t(3)-1) \left(-t(1) t(3)^2+t(2) t(3)^2-t(3)^2-t(1) t(3)-t(2) t(3)+t(1)-t(1) t(2)-t(2)\right)}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{3/2}} (db)
Jones polynomial q^3+3-2 q^{-1} +3 q^{-2} -3 q^{-3} +3 q^{-4} -2 q^{-5} + q^{-6} (db)
Signature 0 (db)
HOMFLY-PT polynomial a^6-2 z^2 a^4-2 a^4+z^4 a^2+3 z^2 a^2+a^2 z^{-2} +4 a^2-z^4-6 z^2-2 z^{-2} -6+z^2 a^{-2} + a^{-2} z^{-2} +3 a^{-2} (db)
Kauffman polynomial a^4 z^8+a^2 z^8+2 a^5 z^7+3 a^3 z^7+a z^7+a^6 z^6-2 a^4 z^6-4 a^2 z^6+z^6 a^{-2} -8 a^5 z^5-13 a^3 z^5-5 a z^5-4 a^6 z^4-4 a^4 z^4+3 a^2 z^4-6 z^4 a^{-2} -3 z^4+7 a^5 z^3+15 a^3 z^3+5 a z^3-3 z^3 a^{-1} +4 a^6 z^2+4 a^4 z^2+a^2 z^2+9 z^2 a^{-2} +10 z^2-2 a^5 z-6 a^3 z+2 a z+6 z a^{-1} -a^6-2 a^2-5 a^{-2} -7-2 a z^{-1} -2 a^{-1} z^{-1} +a^2 z^{-2} + a^{-2} z^{-2} +2 z^{-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
7          11
5          11
3       11  0
1      3    3
-1     241   1
-3    21     1
-5   121     0
-7  22       0
-9 12        1
-11 1         -1
-131          1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-3 i=-1 i=1
r=-6 {\mathbb Z}
r=-5 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=-3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-1 {\mathbb Z} {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=0 {\mathbb Z}_2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}^{3}
r=1 {\mathbb Z} {\mathbb Z}_2 {\mathbb Z}
r=2 {\mathbb Z}_2 {\mathbb Z}
r=3
r=4 {\mathbb Z} {\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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See/edit the Link Page master template (intermediate).

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