L11n99

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L11n98.gif

L11n98

L11n100.gif

L11n100

Contents

L11n99.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n99 at Knotilus!


Link Presentations

[edit Notes on L11n99's Link Presentations]

Planar diagram presentation X6172 X12,3,13,4 X13,21,14,20 X7,17,8,16 X19,9,20,8 X9,19,10,18 X17,11,18,10 X15,5,16,22 X21,15,22,14 X2536 X4,11,1,12
Gauss code {1, -10, 2, -11}, {10, -1, -4, 5, -6, 7, 11, -2, -3, 9, -8, 4, -7, 6, -5, 3, -9, 8}
A Braid Representative
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A Morse Link Presentation L11n99 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{u v^3-u v^2-u v+2 u+2 v^5-v^4-v^3+v^2}{\sqrt{u} v^{5/2}} (db)
Jones polynomial q^{15/2}-q^{13/2}+q^{11/2}-2 q^{3/2}+\sqrt{q}-\frac{2}{\sqrt{q}}+\frac{1}{q^{3/2}}-\frac{1}{q^{5/2}} (db)
Signature 3 (db)
HOMFLY-PT polynomial -z^5 a^{-1} -z^5 a^{-3} +a z^3-4 z^3 a^{-1} -5 z^3 a^{-3} +3 a z-2 z a^{-1} -6 z a^{-3} +2 z a^{-5} +z a^{-7} +a z^{-1} + a^{-1} z^{-1} -4 a^{-3} z^{-1} +2 a^{-5} z^{-1} (db)
Kauffman polynomial -z^9 a^{-1} -z^9 a^{-3} -2 z^8 a^{-2} -z^8 a^{-4} -z^8-a z^7+6 z^7 a^{-1} +8 z^7 a^{-3} -z^7 a^{-7} +13 z^6 a^{-2} +8 z^6 a^{-4} -z^6 a^{-6} -z^6 a^{-8} +5 z^6+6 a z^5-11 z^5 a^{-1} -22 z^5 a^{-3} +5 z^5 a^{-7} -25 z^4 a^{-2} -19 z^4 a^{-4} +6 z^4 a^{-6} +5 z^4 a^{-8} -5 z^4-10 a z^3+10 z^3 a^{-1} +29 z^3 a^{-3} +4 z^3 a^{-5} -5 z^3 a^{-7} +19 z^2 a^{-2} +19 z^2 a^{-4} -7 z^2 a^{-6} -6 z^2 a^{-8} -z^2+5 a z-5 z a^{-1} -16 z a^{-3} -6 z a^{-5} -5 a^{-2} -6 a^{-4} + a^{-6} +2 a^{-8} +1-a z^{-1} + a^{-1} z^{-1} +4 a^{-3} z^{-1} +2 a^{-5} z^{-1} (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 \
-4-3-2-101234567χ
16           1-1
14            0
12         11 0
10       21   -1
8       11   0
6     231    0
4    1 1     2
2   131      1
0  111       1
-2  1         1
-411          0
-61           1
Integral Khovanov Homology

(db, data source)

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

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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L11n98

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L11n100