L11n29

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L11n28.gif

L11n28

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L11n30

Contents

L11n29.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11n29's Link Presentations]

Planar diagram presentation X6172 X18,7,19,8 X4,19,1,20 X5,12,6,13 X3849 X9,16,10,17 X13,22,14,5 X15,10,16,11 X21,14,22,15 X11,20,12,21 X17,2,18,3
Gauss code {1, 11, -5, -3}, {-4, -1, 2, 5, -6, 8, -10, 4, -7, 9, -8, 6, -11, -2, 3, 10, -9, 7}
A Braid Representative
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A Morse Link Presentation L11n29 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{4 t(1) t(2)^3-5 t(1) t(2)^2+2 t(2)^2+2 t(1) t(2)-5 t(2)+4}{\sqrt{t(1)} t(2)^{3/2}} (db)
Jones polynomial -\frac{4}{q^{9/2}}+\frac{2}{q^{7/2}}-\frac{1}{q^{5/2}}+\frac{2}{q^{23/2}}-\frac{3}{q^{21/2}}+\frac{5}{q^{19/2}}-\frac{7}{q^{17/2}}+\frac{7}{q^{15/2}}-\frac{8}{q^{13/2}}+\frac{5}{q^{11/2}} (db)
Signature -5 (db)
HOMFLY-PT polynomial -a^{13} z^{-1} +a^{11} z^3+3 a^{11} z+a^{11} z^{-1} -a^9 z^5-2 a^9 z^3+2 a^9 z+2 a^9 z^{-1} -2 a^7 z^5-7 a^7 z^3-6 a^7 z-2 a^7 z^{-1} -a^5 z^5-3 a^5 z^3-a^5 z (db)
Kauffman polynomial -3 z^4 a^{14}+8 z^2 a^{14}-4 a^{14}-z^7 a^{13}+z^5 a^{13}+z^3 a^{13}+a^{13} z^{-1} -2 z^8 a^{12}+9 z^6 a^{12}-23 z^4 a^{12}+26 z^2 a^{12}-9 a^{12}-z^9 a^{11}+2 z^7 a^{11}-2 z^5 a^{11}-3 z^3 a^{11}+2 z a^{11}+a^{11} z^{-1} -4 z^8 a^{10}+15 z^6 a^{10}-25 z^4 a^{10}+14 z^2 a^{10}-4 a^{10}-z^9 a^9+8 z^5 a^9-19 z^3 a^9+11 z a^9-2 a^9 z^{-1} -2 z^8 a^8+4 z^6 a^8-5 z^2 a^8+2 a^8-3 z^7 a^7+10 z^5 a^7-12 z^3 a^7+8 z a^7-2 a^7 z^{-1} -2 z^6 a^6+5 z^4 a^6-z^2 a^6-z^5 a^5+3 z^3 a^5-z a^5 (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     52   3
-14    34    1
-16   44     0
-18  13      2
-20 24       -2
-22 1        1
-242         -2
Integral Khovanov Homology

(db, data source)

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