L11n100

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L11n99.gif

L11n99

L11n101.gif

L11n101

Contents

L11n100.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n100 at Knotilus!


Link Presentations

[edit Notes on L11n100's Link Presentations]

Planar diagram presentation X6172 X12,3,13,4 X20,13,21,14 X7,17,8,16 X19,9,20,8 X9,19,10,18 X17,11,18,10 X22,15,5,16 X14,21,15,22 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 L11n100 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{2 u v^3-3 u v^2+u v-u-v^3+v^2-3 v+2}{\sqrt{u} v^{3/2}} (db)
Jones polynomial \frac{3}{q^{9/2}}-\frac{4}{q^{7/2}}-q^{5/2}+\frac{4}{q^{5/2}}+2 q^{3/2}-\frac{5}{q^{3/2}}-\frac{2}{q^{11/2}}-3 \sqrt{q}+\frac{4}{\sqrt{q}} (db)
Signature -3 (db)
HOMFLY-PT polynomial a^7 z^{-1} -a^5 z^3-3 a^5 z-2 a^5 z^{-1} +a^3 z^5+3 a^3 z^3+2 a^3 z+2 a^3 z^{-1} +a z^5+3 a z^3-z^3 a^{-1} +a z-a z^{-1} -2 z a^{-1} (db)
Kauffman polynomial -a^3 z^9-a z^9-a^4 z^8-3 a^2 z^8-2 z^8+4 a^3 z^7+3 a z^7-z^7 a^{-1} +2 a^4 z^6+12 a^2 z^6+10 z^6-4 a^5 z^5-9 a^3 z^5+5 z^5 a^{-1} -a^6 z^4+a^4 z^4-12 a^2 z^4-14 z^4+11 a^5 z^3+17 a^3 z^3-a z^3-7 z^3 a^{-1} -a^6 z^2-2 a^4 z^2+4 a^2 z^2+5 z^2-3 a^7 z-9 a^5 z-10 a^3 z-2 a z+2 z a^{-1} +a^4+a^7 z^{-1} +2 a^5 z^{-1} +2 a^3 z^{-1} +a 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 \
-4-3-2-101234χ
6        11
4       1 -1
2      21 1
0     21  -1
-2    32   1
-4   23    1
-6  22     0
-8 12      1
-1012       -1
-122        2
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-4 i=-2
r=-4 {\mathbb Z}^{2} {\mathbb Z}
r=-3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=0 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=2 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=4 {\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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L11n99

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L11n101