L11n355

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L11n354.gif

L11n354

L11n356.gif

L11n356

Contents

L11n355.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11n355's Link Presentations]

Planar diagram presentation X6172 X10,3,11,4 X18,12,19,11 X7,16,8,17 X15,8,16,9 X20,13,21,14 X22,20,15,19 X12,21,13,22 X14,18,5,17 X2536 X4,9,1,10
Gauss code {1, -10, 2, -11}, {-5, 4, 9, -3, 7, -6, 8, -7}, {10, -1, -4, 5, 11, -2, 3, -8, 6, -9}
A Braid Representative
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A Morse Link Presentation L11n355 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(w-1) \left(u v^2 w-u v^2-2 u v w+3 u v+u w-u-v^2 w+v^2+3 v w-2 v-w+1\right)}{\sqrt{u} v w} (db)
Jones polynomial 2 q^{-7} -4 q^{-6} +9 q^{-5} -10 q^{-4} +13 q^{-3} -q^2-12 q^{-2} +4 q+10 q^{-1} -7 (db)
Signature -2 (db)
HOMFLY-PT polynomial a^8 z^{-2} +z^2 a^6-2 a^6 z^{-2} -2 a^6-2 z^4 a^4-2 z^2 a^4+a^4 z^{-2} +a^4+z^6 a^2+3 z^4 a^2+4 z^2 a^2+a^2-z^4-z^2 (db)
Kauffman polynomial 3 a^8 z^4-6 a^8 z^2-a^8 z^{-2} +4 a^8+a^7 z^7+a^7 z^5+a^7 z^3-5 a^7 z+2 a^7 z^{-1} +2 a^6 z^8-2 a^6 z^6+7 a^6 z^4-10 a^6 z^2-2 a^6 z^{-2} +6 a^6+a^5 z^9+4 a^5 z^7-7 a^5 z^5+7 a^5 z^3-5 a^5 z+2 a^5 z^{-1} +6 a^4 z^8-7 a^4 z^6+2 a^4 z^4-a^4 z^2-a^4 z^{-2} +2 a^4+a^3 z^9+9 a^3 z^7-20 a^3 z^5+11 a^3 z^3+4 a^2 z^8-a^2 z^6-9 a^2 z^4+5 a^2 z^2-a^2+6 a z^7-11 a z^5+z^5 a^{-1} +4 a z^3-z^3 a^{-1} +4 z^6-7 z^4+2 z^2 (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 \
-6-5-4-3-2-10123χ
5         1-1
3        3 3
1       41 -3
-1      63  3
-3     75   -2
-5    65    1
-7   58     3
-9  45      -1
-11 16       5
-1313        -2
-152         2
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-3 i=-1
r=-6 {\mathbb Z}^{2} {\mathbb Z}
r=-5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{4}
r=-3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{6}
r=-1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=0 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{6}
r=1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=2 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=3 {\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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L11n354

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L11n356