L11n237

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L11n236.gif

L11n236

L11n238.gif

L11n238

Contents

L11n237.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n237 at Knotilus!


Link Presentations

[edit Notes on L11n237's Link Presentations]

Planar diagram presentation X10,1,11,2 X3,14,4,15 X22,5,9,6 X6,9,7,10 X20,12,21,11 X18,14,19,13 X12,20,13,19 X7,16,8,17 X15,4,16,5 X17,8,18,1 X2,21,3,22
Gauss code {1, -11, -2, 9, 3, -4, -8, 10}, {4, -1, 5, -7, 6, 2, -9, 8, -10, -6, 7, -5, 11, -3}
A Braid Representative
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A Morse Link Presentation L11n237 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{\left(t(2) t(1)^2-2 t(1)^2-2 t(2)+1\right) \left(t(1) t(2)^2+1\right)}{t(1)^{3/2} t(2)^{3/2}} (db)
Jones polynomial -\frac{1}{\sqrt{q}}+\frac{1}{q^{3/2}}-\frac{1}{q^{5/2}}+\frac{1}{q^{7/2}}-\frac{1}{q^{11/2}}-\frac{1}{q^{15/2}}+\frac{1}{q^{17/2}}-\frac{1}{q^{19/2}} (db)
Signature -4 (db)
HOMFLY-PT polynomial 2 a^9 z+a^9 z^{-1} -a^7 z^5-6 a^7 z^3-7 a^7 z-a^7 z^{-1} +a^5 z^7+6 a^5 z^5+10 a^5 z^3+6 a^5 z-a^3 z^5-5 a^3 z^3-5 a^3 z (db)
Kauffman polynomial -z^5 a^{11}+4 z^3 a^{11}-2 z a^{11}-z^6 a^{10}+4 z^4 a^{10}-2 z^2 a^{10}-z^7 a^9+6 z^5 a^9-10 z^3 a^9+7 z a^9-a^9 z^{-1} -z^6 a^8+6 z^4 a^8-6 z^2 a^8+a^8-2 z^7 a^7+13 z^5 a^7-21 z^3 a^7+9 z a^7-a^7 z^{-1} -z^8 a^6+6 z^6 a^6-8 z^4 a^6+2 z^2 a^6-2 z^7 a^5+12 z^5 a^5-17 z^3 a^5+5 z a^5-z^8 a^4+6 z^6 a^4-10 z^4 a^4+6 z^2 a^4-z^7 a^3+6 z^5 a^3-10 z^3 a^3+5 z a^3 (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 \
-8-7-6-5-4-3-2-1012χ
0          11
-2           0
-4       121 0
-6       11  0
-8     221   -1
-10    111    1
-12   131     1
-14  111      1
-16  11       0
-1811         0
-201          1
Integral Khovanov Homology

(db, data source)

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

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L11n236

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L11n238