L11a156

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L11a155.gif

L11a155

L11a157.gif

L11a157

Contents

L11a156.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a156 at Knotilus!


Link Presentations

[edit Notes on L11a156's Link Presentations]

Planar diagram presentation X8192 X10,4,11,3 X22,10,7,9 X14,6,15,5 X18,14,19,13 X20,17,21,18 X16,21,17,22 X12,20,13,19 X2738 X4,12,5,11 X6,16,1,15
Gauss code {1, -9, 2, -10, 4, -11}, {9, -1, 3, -2, 10, -8, 5, -4, 11, -7, 6, -5, 8, -6, 7, -3}
A Braid Representative
BraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gif
BraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11a156 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{u^2 v^4-5 u^2 v^3+6 u^2 v^2-2 u^2 v-u v^4+6 u v^3-9 u v^2+6 u v-u-2 v^3+6 v^2-5 v+1}{u v^2} (db)
Jones polynomial q^{17/2}-3 q^{15/2}+6 q^{13/2}-11 q^{11/2}+14 q^{9/2}-16 q^{7/2}+16 q^{5/2}-14 q^{3/2}+10 \sqrt{q}-\frac{7}{\sqrt{q}}+\frac{3}{q^{3/2}}-\frac{1}{q^{5/2}} (db)
Signature 3 (db)
HOMFLY-PT polynomial z^3 a^{-7} +2 z a^{-7} -2 z^5 a^{-5} -6 z^3 a^{-5} -5 z a^{-5} - a^{-5} z^{-1} +z^7 a^{-3} +4 z^5 a^{-3} +7 z^3 a^{-3} +7 z a^{-3} +2 a^{-3} z^{-1} -2 z^5 a^{-1} +a z^3-6 z^3 a^{-1} +2 a z-5 z a^{-1} +a z^{-1} -2 a^{-1} z^{-1} (db)
Kauffman polynomial -z^{10} a^{-2} -z^{10} a^{-4} -3 z^9 a^{-1} -7 z^9 a^{-3} -4 z^9 a^{-5} -8 z^8 a^{-2} -12 z^8 a^{-4} -7 z^8 a^{-6} -3 z^8-a z^7+5 z^7 a^{-1} +10 z^7 a^{-3} -3 z^7 a^{-5} -7 z^7 a^{-7} +35 z^6 a^{-2} +39 z^6 a^{-4} +10 z^6 a^{-6} -5 z^6 a^{-8} +11 z^6+4 a z^5+11 z^5 a^{-1} +20 z^5 a^{-3} +25 z^5 a^{-5} +9 z^5 a^{-7} -3 z^5 a^{-9} -35 z^4 a^{-2} -34 z^4 a^{-4} -6 z^4 a^{-6} +4 z^4 a^{-8} -z^4 a^{-10} -12 z^4-6 a z^3-23 z^3 a^{-1} -34 z^3 a^{-3} -26 z^3 a^{-5} -6 z^3 a^{-7} +3 z^3 a^{-9} +11 z^2 a^{-2} +11 z^2 a^{-4} +2 z^2 a^{-6} -z^2 a^{-8} +z^2 a^{-10} +4 z^2+4 a z+12 z a^{-1} +15 z a^{-3} +9 z a^{-5} +z a^{-7} -z a^{-9} - 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 \
-4-3-2-101234567χ
18           1-1
16          2 2
14         41 -3
12        72  5
10       74   -3
8      97    2
6     88     0
4    68      -2
2   59       4
0  25        -3
-2 15         4
-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}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=0 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{6}
r=1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{9}
r=3 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} {\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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L11a155

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L11a157