L11n201

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L11n200.gif

L11n200

L11n202.gif

L11n202

Contents

L11n201.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n201 at Knotilus!


Link Presentations

[edit Notes on L11n201's Link Presentations]

Planar diagram presentation X10,1,11,2 X12,3,13,4 X5,14,6,15 X16,7,17,8 X15,21,16,20 X18,14,19,13 X21,6,22,7 X22,18,9,17 X4,19,5,20 X2,9,3,10 X8,11,1,12
Gauss code {1, -10, 2, -9, -3, 7, 4, -11}, {10, -1, 11, -2, 6, 3, -5, -4, 8, -6, 9, 5, -7, -8}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart0.gif
BraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gif
A Morse Link Presentation L11n201 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{u^3 v+u^2 v^4-u^2 v^3+2 u^2 v^2-2 u^2 v+u^2+u v^5-2 u v^4+2 u v^3-u v^2+u v+v^4}{u^{3/2} v^{5/2}} (db)
Jones polynomial \sqrt{q}-\frac{3}{\sqrt{q}}+\frac{4}{q^{3/2}}-\frac{5}{q^{5/2}}+\frac{5}{q^{7/2}}-\frac{5}{q^{9/2}}+\frac{4}{q^{11/2}}-\frac{3}{q^{13/2}}+\frac{1}{q^{15/2}}-\frac{1}{q^{17/2}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a^7 z^3+3 a^7 z+2 a^7 z^{-1} -a^5 z^5-4 a^5 z^3-6 a^5 z-3 a^5 z^{-1} -a^3 z^5-3 a^3 z^3-2 a^3 z+a^3 z^{-1} +a z^3+a z (db)
Kauffman polynomial -z^7 a^9+6 z^5 a^9-11 z^3 a^9+6 z a^9-z^8 a^8+4 z^6 a^8-3 z^4 a^8-z^2 a^8-z^9 a^7+4 z^7 a^7-5 z^5 a^7+6 z^3 a^7-6 z a^7+2 a^7 z^{-1} -3 z^8 a^6+12 z^6 a^6-14 z^4 a^6+9 z^2 a^6-3 a^6-z^9 a^5+3 z^7 a^5-6 z^5 a^5+15 z^3 a^5-12 z a^5+3 a^5 z^{-1} -2 z^8 a^4+7 z^6 a^4-11 z^4 a^4+11 z^2 a^4-3 a^4-2 z^7 a^3+5 z^5 a^3-5 z^3 a^3+z a^3+a^3 z^{-1} -z^6 a^2-a^2-3 z^3 a+z a-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 \
-8-7-6-5-4-3-2-101χ
2         1-1
0        2 2
-2       32 -1
-4      21  1
-6     33   0
-8    22    0
-10   23     1
-12  12      -1
-14  2       2
-1611        0
-181         1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-2 i=0
r=-8 {\mathbb Z} {\mathbb Z}
r=-7 {\mathbb Z}
r=-6 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-1 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=0 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=1 {\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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L11n200

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L11n202