L10a80

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L10a79.gif

L10a79

L10a81.gif

L10a81

Contents

L10a80.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L10a80's Link Presentations]

Planar diagram presentation X8192 X12,3,13,4 X18,10,19,9 X20,13,7,14 X14,19,15,20 X10,16,11,15 X16,6,17,5 X2738 X4,11,5,12 X6,18,1,17
Gauss code {1, -8, 2, -9, 7, -10}, {8, -1, 3, -6, 9, -2, 4, -5, 6, -7, 10, -3, 5, -4}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
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BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
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A Morse Link Presentation L10a80 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{2 u^2 v^3-4 u^2 v^2+3 u^2 v-u^2+u v^4-4 u v^3+5 u v^2-4 u v+u-v^4+3 v^3-4 v^2+2 v}{u v^2} (db)
Jones polynomial -q^{9/2}+3 q^{7/2}-6 q^{5/2}+9 q^{3/2}-11 \sqrt{q}+\frac{11}{\sqrt{q}}-\frac{11}{q^{3/2}}+\frac{8}{q^{5/2}}-\frac{6}{q^{7/2}}+\frac{3}{q^{9/2}}-\frac{1}{q^{11/2}} (db)
Signature 1 (db)
HOMFLY-PT polynomial a^5 z+a^5 z^{-1} -3 a^3 z^3-z^3 a^{-3} -6 a^3 z-2 a^3 z^{-1} -z a^{-3} +2 a z^5+z^5 a^{-1} +6 a z^3+z^3 a^{-1} +6 a z+2 a z^{-1} -z a^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial -2 a^3 z^9-2 a z^9-3 a^4 z^8-10 a^2 z^8-7 z^8-a^5 z^7+a^3 z^7-8 a z^7-10 z^7 a^{-1} +12 a^4 z^6+34 a^2 z^6-9 z^6 a^{-2} +13 z^6+4 a^5 z^5+19 a^3 z^5+42 a z^5+21 z^5 a^{-1} -6 z^5 a^{-3} -14 a^4 z^4-30 a^2 z^4+13 z^4 a^{-2} -3 z^4 a^{-4} -6 a^5 z^3-32 a^3 z^3-44 a z^3-13 z^3 a^{-1} +4 z^3 a^{-3} -z^3 a^{-5} +4 a^4 z^2+7 a^2 z^2-5 z^2 a^{-2} -2 z^2+4 a^5 z+15 a^3 z+17 a z+5 z a^{-1} -z a^{-3} -a^2-a^5 z^{-1} -2 a^3 z^{-1} -2 a z^{-1} - a^{-1} 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 \
-6-5-4-3-2-101234χ
10          11
8         2 -2
6        41 3
4       52  -3
2      64   2
0     66    0
-2    55     0
-4   47      3
-6  24       -2
-8 14        3
-10 2         -2
-121          1
Integral Khovanov Homology

(db, data source)

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