L11n26

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L11n25

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L11n27

Contents

L11n26.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11n26's Link Presentations]

Planar diagram presentation X6172 X16,7,17,8 X4,17,1,18 X5,12,6,13 X3849 X13,22,14,5 X21,14,22,15 X11,18,12,19 X9,20,10,21 X19,10,20,11 X15,2,16,3
Gauss code {1, 11, -5, -3}, {-4, -1, 2, 5, -9, 10, -8, 4, -6, 7, -11, -2, 3, 8, -10, 9, -7, 6}
A Braid Representative
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A Morse Link Presentation L11n26 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{4 t(1) t(2)^3-4 t(1) t(2)^2+t(2)^2+t(1) t(2)-4 t(2)+4}{\sqrt{t(1)} t(2)^{3/2}} (db)
Jones polynomial -\frac{1}{q^{5/2}}+\frac{2}{q^{7/2}}-\frac{4}{q^{9/2}}+\frac{5}{q^{11/2}}-\frac{7}{q^{13/2}}+\frac{5}{q^{15/2}}-\frac{6}{q^{17/2}}+\frac{4}{q^{19/2}}-\frac{1}{q^{21/2}}+\frac{1}{q^{23/2}} (db)
Signature -5 (db)
HOMFLY-PT polynomial -a^{13} z^{-1} +z^3 a^{11}+4 z a^{11}+a^{11} z^{-1} -z^5 a^9-3 z^3 a^9+2 a^9 z^{-1} -2 z^5 a^7-7 z^3 a^7-5 z a^7-2 a^7 z^{-1} -z^5 a^5-3 z^3 a^5-z a^5 (db)
Kauffman polynomial a^{14} z^4-4 a^{14} z^2+4 a^{14}+a^{13} z^5-a^{13} z^3-a^{13} z-a^{13} z^{-1} +a^{12} z^8-6 a^{12} z^6+18 a^{12} z^4-21 a^{12} z^2+9 a^{12}+a^{11} z^9-5 a^{11} z^7+11 a^{11} z^5-4 a^{11} z^3-3 a^{11} z-a^{11} z^{-1} +3 a^{10} z^8-13 a^{10} z^6+24 a^{10} z^4-14 a^{10} z^2+4 a^{10}+a^9 z^9-2 a^9 z^7-a^9 z^5+11 a^9 z^3-10 a^9 z+2 a^9 z^{-1} +2 a^8 z^8-5 a^8 z^6+2 a^8 z^4+4 a^8 z^2-2 a^8+3 a^7 z^7-10 a^7 z^5+11 a^7 z^3-7 a^7 z+2 a^7 z^{-1} +2 a^6 z^6-5 a^6 z^4+a^6 z^2+a^5 z^5-3 a^5 z^3+a^5 z (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 \
-9-8-7-6-5-4-3-2-10χ
-4         11
-6        21-1
-8       2  2
-10      32  -1
-12     42   2
-14    24    2
-16   43     1
-18   2      2
-20 14       -3
-22          0
-241         -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-6 i=-4
r=-9 {\mathbb Z}
r=-8 {\mathbb Z}_2 {\mathbb Z}
r=-7 {\mathbb Z}^{4}
r=-6 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{4}
r=-3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
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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L11n25

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