L11n158

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L11n157.gif

L11n157

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L11n159

Contents

L11n158.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11n158's Link Presentations]

Planar diagram presentation X8192 X16,7,17,8 X3,10,4,11 X2,15,3,16 X14,10,15,9 X11,19,12,18 X12,5,13,6 X6,21,1,22 X20,14,21,13 X22,17,7,18 X19,4,20,5
Gauss code {1, -4, -3, 11, 7, -8}, {2, -1, 5, 3, -6, -7, 9, -5, 4, -2, 10, 6, -11, -9, 8, -10}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11n158 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{2 u^2 v-u^2+u v^4-u v^3+u v^2-u v+u-v^4+2 v^3}{u v^2} (db)
Jones polynomial -\frac{4}{q^{9/2}}+\frac{3}{q^{7/2}}-\frac{3}{q^{5/2}}+\frac{2}{q^{3/2}}-\frac{1}{q^{17/2}}+\frac{2}{q^{15/2}}-\frac{3}{q^{13/2}}+\frac{3}{q^{11/2}}-\frac{1}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a^7 z^3+2 a^7 z+a^7 z^{-1} -a^5 z^5-4 a^5 z^3-5 a^5 z-a^5 z^{-1} +a^3 z^3+a^3 z-a z (db)
Kauffman polynomial -z^7 a^9+5 z^5 a^9-7 z^3 a^9+3 z a^9-2 z^8 a^8+10 z^6 a^8-13 z^4 a^8+4 z^2 a^8-z^9 a^7+3 z^7 a^7+2 z^5 a^7-5 z^3 a^7-z a^7+a^7 z^{-1} -3 z^8 a^6+14 z^6 a^6-17 z^4 a^6+7 z^2 a^6-a^6-z^9 a^5+4 z^7 a^5-4 z^5 a^5+4 z^3 a^5-3 z a^5+a^5 z^{-1} -z^8 a^4+4 z^6 a^4-4 z^4 a^4+2 z^2 a^4-z^5 a^3+2 z^3 a^3-z^2 a^2-z a (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 \
-8-7-6-5-4-3-2-10χ
0        11
-2       21-1
-4      1  1
-6     22  0
-8    21   1
-10   12    1
-12  22     0
-14 12      1
-16 1       -1
-181        1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-2 i=0
r=-8 {\mathbb Z}
r=-7 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-6 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=-5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-3 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=0 {\mathbb Z} {\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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L11n157.gif

L11n157

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L11n159