L11a177

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L11a176.gif

L11a176

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L11a178

Contents

L11a177.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a177 at Knotilus!


Link Presentations

[edit Notes on L11a177's Link Presentations]

Planar diagram presentation X8192 X10,4,11,3 X16,7,17,8 X18,10,19,9 X20,11,21,12 X22,15,7,16 X12,21,13,22 X14,6,15,5 X4,14,5,13 X6,18,1,17 X2,19,3,20
Gauss code {1, -11, 2, -9, 8, -10}, {3, -1, 4, -2, 5, -7, 9, -8, 6, -3, 10, -4, 11, -5, 7, -6}
A Braid Representative
BraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart4.gif
A Morse Link Presentation L11a177 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{\left(v^2-v+1\right) \left(u v^2-u v+u+v-1\right) \left(u v^2-u v-v^2+v-1\right)}{u v^3} (db)
Jones polynomial q^{9/2}-\frac{9}{q^{9/2}}-4 q^{7/2}+\frac{16}{q^{7/2}}+9 q^{5/2}-\frac{21}{q^{5/2}}-16 q^{3/2}+\frac{24}{q^{3/2}}-\frac{1}{q^{13/2}}+\frac{4}{q^{11/2}}+20 \sqrt{q}-\frac{25}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a^3 z^7+4 a^3 z^5+5 a^3 z^3+a^3 z-a^3 z^{-1} -a z^9-6 a z^7+z^7 a^{-1} -13 a z^5+4 z^5 a^{-1} -11 a z^3+5 z^3 a^{-1} -a z+z a^{-1} +3 a z^{-1} -2 a^{-1} z^{-1} (db)
Kauffman polynomial -3 a^2 z^{10}-3 z^{10}-8 a^3 z^9-15 a z^9-7 z^9 a^{-1} -10 a^4 z^8-11 a^2 z^8-7 z^8 a^{-2} -8 z^8-8 a^5 z^7+6 a^3 z^7+28 a z^7+10 z^7 a^{-1} -4 z^7 a^{-3} -4 a^6 z^6+13 a^4 z^6+26 a^2 z^6+14 z^6 a^{-2} -z^6 a^{-4} +24 z^6-a^7 z^5+11 a^5 z^5+a^3 z^5-20 a z^5+9 z^5 a^{-3} +5 a^6 z^4-5 a^4 z^4-14 a^2 z^4-6 z^4 a^{-2} +2 z^4 a^{-4} -12 z^4+a^7 z^3-5 a^5 z^3+2 a^3 z^3+12 a z^3-2 z^3 a^{-1} -6 z^3 a^{-3} -2 a^6 z^2-a^4 z^2-z^2 a^{-2} -z^2 a^{-4} -z^2-2 a^3 z-a z+2 z a^{-1} +z a^{-3} +a^4+3 a^2+3-a^3 z^{-1} -3 a z^{-1} -2 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 \
-6-5-4-3-2-1012345χ
10           1-1
8          3 3
6         61 -5
4        103  7
2       117   -4
0      149    5
-2     1112     1
-4    1013      -3
-6   611       5
-8  310        -7
-10 16         5
-12 3          -3
-141           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-2 i=0
r=-6 {\mathbb Z}
r=-5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-3 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-2 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=-1 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=0 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{14}
r=1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{10}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=5 {\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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