L10a86

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L10a85.gif

L10a85

L10a87.gif

L10a87

Contents

L10a86.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L10a86's Link Presentations]

Planar diagram presentation X8192 X16,6,17,5 X18,10,19,9 X14,19,15,20 X10,16,11,15 X20,11,7,12 X4758 X2,14,3,13 X12,4,13,3 X6,18,1,17
Gauss code {1, -8, 9, -7, 2, -10}, {7, -1, 3, -5, 6, -9, 8, -4, 5, -2, 10, -3, 4, -6}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gif
BraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gif
BraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L10a86 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{u^2 v^4-3 u^2 v^3+4 u^2 v^2-3 u^2 v+u^2-2 u v^4+7 u v^3-11 u v^2+7 u v-2 u+v^4-3 v^3+4 v^2-3 v+1}{u v^2} (db)
Jones polynomial -9 q^{9/2}+13 q^{7/2}-\frac{1}{q^{7/2}}-17 q^{5/2}+\frac{4}{q^{5/2}}+18 q^{3/2}-\frac{9}{q^{3/2}}-q^{13/2}+4 q^{11/2}-17 \sqrt{q}+\frac{13}{\sqrt{q}} (db)
Signature 1 (db)
HOMFLY-PT polynomial -z^7 a^{-1} +a z^5-4 z^5 a^{-1} +2 z^5 a^{-3} +2 a z^3-7 z^3 a^{-1} +5 z^3 a^{-3} -z^3 a^{-5} +2 a z-5 z a^{-1} +5 z a^{-3} -z a^{-5} +a z^{-1} -2 a^{-1} z^{-1} +2 a^{-3} z^{-1} - a^{-5} z^{-1} (db)
Kauffman polynomial -3 z^9 a^{-1} -3 z^9 a^{-3} -16 z^8 a^{-2} -8 z^8 a^{-4} -8 z^8-8 a z^7-12 z^7 a^{-1} -12 z^7 a^{-3} -8 z^7 a^{-5} -4 a^2 z^6+30 z^6 a^{-2} +11 z^6 a^{-4} -4 z^6 a^{-6} +11 z^6-a^3 z^5+14 a z^5+36 z^5 a^{-1} +36 z^5 a^{-3} +14 z^5 a^{-5} -z^5 a^{-7} +5 a^2 z^4-18 z^4 a^{-2} -4 z^4 a^{-4} +5 z^4 a^{-6} -4 z^4+a^3 z^3-9 a z^3-31 z^3 a^{-1} -31 z^3 a^{-3} -9 z^3 a^{-5} +z^3 a^{-7} -a^2 z^2+4 z^2 a^{-2} +z^2 a^{-4} -z^2 a^{-6} +z^2+4 a z+12 z a^{-1} +12 z a^{-3} +4 z a^{-5} - a^{-2} -a z^{-1} -2 a^{-1} z^{-1} -2 a^{-3} z^{-1} - a^{-5} 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 \
-4-3-2-10123456χ
14          11
12         3 -3
10        61 5
8       73  -4
6      106   4
4     98    -1
2    89     -1
0   610      4
-2  37       -4
-4 16        5
-6 3         -3
-81          1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=0 i=2
r=-4 {\mathbb Z}
r=-3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=0 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{8}
r=1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{10}
r=3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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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L10a85

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L10a87