L11n221

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L11n220.gif

L11n220

L11n222.gif

L11n222

Contents

L11n221.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n221 at Knotilus!


Link Presentations

[edit Notes on L11n221's Link Presentations]

Planar diagram presentation X10,1,11,2 X8,9,1,10 X12,4,13,3 X22,16,9,15 X2,17,3,18 X21,4,22,5 X5,15,6,14 X13,21,14,20 X16,12,17,11 X6,19,7,20 X18,7,19,8
Gauss code {1, -5, 3, 6, -7, -10, 11, -2}, {2, -1, 9, -3, -8, 7, 4, -9, 5, -11, 10, 8, -6, -4}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
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BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart4.gifBraidPart0.gifBraidPart4.gif
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A Morse Link Presentation L11n221 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{t(2)^2 t(1)^3-t(2) t(1)^3+2 t(2)^3 t(1)^2-6 t(2)^2 t(1)^2+6 t(2) t(1)^2-t(1)^2-t(2)^3 t(1)+6 t(2)^2 t(1)-6 t(2) t(1)+2 t(1)-t(2)^2+t(2)}{t(1)^{3/2} t(2)^{3/2}} (db)
Jones polynomial q^{9/2}-\frac{2}{q^{9/2}}-3 q^{7/2}+\frac{5}{q^{7/2}}+6 q^{5/2}-\frac{9}{q^{5/2}}-9 q^{3/2}+\frac{10}{q^{3/2}}+11 \sqrt{q}-\frac{12}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial 2 a^3 z^3+z^3 a^{-3} +3 a^3 z+z a^{-3} +2 a^3 z^{-1} -2 a z^5-z^5 a^{-1} -6 a z^3-z^3 a^{-1} -7 a z+z a^{-1} -3 a z^{-1} + a^{-1} z^{-1} (db)
Kauffman polynomial -2 a z^9-2 z^9 a^{-1} -4 a^2 z^8-4 z^8 a^{-2} -8 z^8-3 a^3 z^7-3 z^7 a^{-3} -a^4 z^6+7 a^2 z^6+11 z^6 a^{-2} -z^6 a^{-4} +20 z^6-a z^5+8 z^5 a^{-1} +9 z^5 a^{-3} -4 a^4 z^4-11 a^2 z^4-8 z^4 a^{-2} +3 z^4 a^{-4} -18 z^4-3 a^5 z^3+3 a^3 z^3+11 a z^3-2 z^3 a^{-1} -7 z^3 a^{-3} +2 a^4 z^2+9 a^2 z^2+3 z^2 a^{-2} -2 z^2 a^{-4} +12 z^2+2 a^5 z-5 a^3 z-10 a z-2 z a^{-1} +z a^{-3} -3 a^2- a^{-2} -3+2 a^3 z^{-1} +3 a z^{-1} + a^{-1} 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 \
-4-3-2-1012345χ
10         1-1
8        2 2
6       41 -3
4      52  3
2     64   -2
0    65    1
-2   57     2
-4  45      -1
-6 15       4
-814        -3
-102         2
Integral Khovanov Homology

(db, data source)

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

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L11n220.gif

L11n220

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L11n222