L10a89

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L10a88.gif

L10a88

L10a90.gif

L10a90

Contents

L10a89.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L10a89 at Knotilus!


Link Presentations

[edit Notes on L10a89's Link Presentations]

Planar diagram presentation X10,1,11,2 X12,4,13,3 X20,12,9,11 X14,6,15,5 X2,9,3,10 X4,14,5,13 X18,16,19,15 X16,7,17,8 X6,17,7,18 X8,20,1,19
Gauss code {1, -5, 2, -6, 4, -9, 8, -10}, {5, -1, 3, -2, 6, -4, 7, -8, 9, -7, 10, -3}
A Braid Representative
BraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gif
BraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart4.gifBraidPart1.gif
BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart0.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L10a89 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(u-1) (v-1) \left(u^2 v^2-u^2 v-u v^2-u-v+1\right)}{u^{3/2} v^{3/2}} (db)
Jones polynomial -q^{15/2}+3 q^{13/2}-4 q^{11/2}+6 q^{9/2}-8 q^{7/2}+7 q^{5/2}-7 q^{3/2}+5 \sqrt{q}-\frac{4}{\sqrt{q}}+\frac{2}{q^{3/2}}-\frac{1}{q^{5/2}} (db)
Signature 3 (db)
HOMFLY-PT polynomial -z^5 a^{-5} -3 z^3 a^{-5} -z a^{-5} +z^7 a^{-3} +5 z^5 a^{-3} +8 z^3 a^{-3} +5 z a^{-3} -2 z^5 a^{-1} +a z^3-8 z^3 a^{-1} +3 a z-7 z a^{-1} +a z^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial z^3 a^{-9} +3 z^4 a^{-8} -2 z^2 a^{-8} +4 z^5 a^{-7} -3 z^3 a^{-7} +z a^{-7} +4 z^6 a^{-6} -2 z^4 a^{-6} -3 z^2 a^{-6} +4 z^7 a^{-5} -5 z^5 a^{-5} -2 z^3 a^{-5} +2 z a^{-5} +3 z^8 a^{-4} -5 z^6 a^{-4} -z^4 a^{-4} +z^2 a^{-4} +z^9 a^{-3} +3 z^7 a^{-3} -18 z^5 a^{-3} +17 z^3 a^{-3} -4 z a^{-3} +5 z^8 a^{-2} -18 z^6 a^{-2} +15 z^4 a^{-2} -z^2 a^{-2} +z^9 a^{-1} +a z^7-5 a z^5-14 z^5 a^{-1} +8 a z^3+23 z^3 a^{-1} -5 a z-10 z a^{-1} +a z^{-1} + a^{-1} z^{-1} +2 z^8-9 z^6+11 z^4-3 z^2-1 (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 \
-4-3-2-10123456χ
16          11
14         2 -2
12        21 1
10       42  -2
8      42   2
6     34    1
4    44     0
2   35      2
0  12       -1
-2 13        2
-4 1         -1
-61          1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=2 i=4
r=-4 {\mathbb Z}
r=-3 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=0 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{4}
r=1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=6 {\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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L10a88

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L10a90