L11n170

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L11n169.gif

L11n169

L11n171.gif

L11n171

Contents

L11n170.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n170 at Knotilus!


Link Presentations

[edit Notes on L11n170's Link Presentations]

Planar diagram presentation X8192 X11,19,12,18 X10,4,11,3 X2,17,3,18 X14,5,15,6 X6718 X16,10,17,9 X13,21,14,20 X19,13,20,12 X22,16,7,15 X4,22,5,21
Gauss code {1, -4, 3, -11, 5, -6}, {6, -1, 7, -3, -2, 9, -8, -5, 10, -7, 4, 2, -9, 8, 11, -10}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart1.gifBraidPart1.gifBraidPart1.gif
BraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart2.gif
A Morse Link Presentation L11n170 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{u^2 v^4-3 u^2 v^3+3 u^2 v^2-u^2 v+3 u v^3-5 u v^2+3 u v-v^3+3 v^2-3 v+1}{u v^2} (db)
Jones polynomial -7 q^{9/2}+9 q^{7/2}-10 q^{5/2}+7 q^{3/2}-\frac{1}{q^{3/2}}+q^{15/2}-3 q^{13/2}+6 q^{11/2}-7 \sqrt{q}+\frac{3}{\sqrt{q}} (db)
Signature 1 (db)
HOMFLY-PT polynomial z^5 a^{-5} +3 z^3 a^{-5} +3 z a^{-5} +2 a^{-5} z^{-1} -z^7 a^{-3} -5 z^5 a^{-3} -9 z^3 a^{-3} -8 z a^{-3} -5 a^{-3} z^{-1} +z^5 a^{-1} +3 z^3 a^{-1} +4 z a^{-1} +3 a^{-1} z^{-1} (db)
Kauffman polynomial -z^9 a^{-3} -z^9 a^{-5} -z^8 a^{-2} -4 z^8 a^{-4} -3 z^8 a^{-6} -3 z^7 a^{-5} -3 z^7 a^{-7} -2 z^6 a^{-2} +5 z^6 a^{-4} +6 z^6 a^{-6} -z^6 a^{-8} -7 z^5 a^{-1} -5 z^5 a^{-3} +11 z^5 a^{-5} +9 z^5 a^{-7} +3 z^4 a^{-4} +3 z^4 a^{-6} +3 z^4 a^{-8} -3 z^4-a z^3+9 z^3 a^{-1} +14 z^3 a^{-3} -2 z^3 a^{-5} -6 z^3 a^{-7} +5 z^2 a^{-2} +2 z^2 a^{-4} -6 z^2 a^{-6} -3 z^2 a^{-8} -7 z a^{-1} -10 z a^{-3} -3 z a^{-5} -5 a^{-2} -5 a^{-4} + a^{-8} +3 a^{-1} z^{-1} +5 a^{-3} z^{-1} +2 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 \
-2-101234567χ
16         1-1
14        2 2
12       41 -3
10      32  1
8     64   -2
6    43    1
4   36     3
2  44      0
0  4       4
-213        -2
-41         1
Integral Khovanov Homology

(db, data source)

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

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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L11n169.gif

L11n169

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L11n171