L11a287

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L11a286.gif

L11a286

L11a288.gif

L11a288

Contents

L11a287.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a287 at Knotilus!


Link Presentations

[edit Notes on L11a287's Link Presentations]

Planar diagram presentation X10,1,11,2 X12,4,13,3 X16,9,17,10 X20,12,21,11 X22,15,9,16 X14,6,15,5 X18,8,19,7 X4,14,5,13 X6,18,7,17 X8,20,1,19 X2,21,3,22
Gauss code {1, -11, 2, -8, 6, -9, 7, -10}, {3, -1, 4, -2, 8, -6, 5, -3, 9, -7, 10, -4, 11, -5}
A Braid Representative
BraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gif
BraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart0.gif
A Morse Link Presentation L11a287 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(u v+1) \left(u v^2-u v+u+v-1\right) \left(u v^2-u v-v^2+v-1\right)}{u^{3/2} v^{5/2}} (db)
Jones polynomial 14 q^{9/2}-16 q^{7/2}+15 q^{5/2}-\frac{1}{q^{5/2}}-14 q^{3/2}+\frac{3}{q^{3/2}}+q^{17/2}-3 q^{15/2}+6 q^{13/2}-11 q^{11/2}+10 \sqrt{q}-\frac{6}{\sqrt{q}} (db)
Signature 3 (db)
HOMFLY-PT polynomial z^7 a^{-5} +5 z^5 a^{-5} +8 z^3 a^{-5} +3 z a^{-5} - a^{-5} z^{-1} -z^9 a^{-3} -7 z^7 a^{-3} -18 z^5 a^{-3} -19 z^3 a^{-3} -5 z a^{-3} + a^{-3} z^{-1} +z^7 a^{-1} +5 z^5 a^{-1} +8 z^3 a^{-1} +4 z a^{-1} (db)
Kauffman polynomial -2 z^{10} a^{-2} -2 z^{10} a^{-4} -4 z^9 a^{-1} -9 z^9 a^{-3} -5 z^9 a^{-5} +z^8 a^{-2} -3 z^8 a^{-4} -7 z^8 a^{-6} -3 z^8-a z^7+15 z^7 a^{-1} +29 z^7 a^{-3} +6 z^7 a^{-5} -7 z^7 a^{-7} +11 z^6 a^{-2} +14 z^6 a^{-4} +10 z^6 a^{-6} -5 z^6 a^{-8} +12 z^6+4 a z^5-19 z^5 a^{-1} -35 z^5 a^{-3} +9 z^5 a^{-7} -3 z^5 a^{-9} -12 z^4 a^{-2} -7 z^4 a^{-4} -4 z^4 a^{-6} +4 z^4 a^{-8} -z^4 a^{-10} -14 z^4-4 a z^3+13 z^3 a^{-1} +26 z^3 a^{-3} +z^3 a^{-5} -5 z^3 a^{-7} +3 z^3 a^{-9} +4 z^2 a^{-2} -2 z^2 a^{-6} -z^2 a^{-8} +z^2 a^{-10} +4 z^2-5 z a^{-1} -5 z a^{-3} -z a^{-7} -z a^{-9} + a^{-4} - 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 \
-4-3-2-101234567χ
18           1-1
16          2 2
14         41 -3
12        72  5
10       85   -3
8      86    2
6     78     1
4    78      -1
2   48       4
0  26        -4
-2 14         3
-4 2          -2
-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}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=0 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{7}
r=1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=4 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{7}
r=5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=6 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=7 {\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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See/edit the Link Page master template (intermediate).

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L11a286

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L11a288