L11n244

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L11n243.gif

L11n243

L11n245.gif

L11n245

Contents

L11n244.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n244 at Knotilus!


Link Presentations

[edit Notes on L11n244's Link Presentations]

Planar diagram presentation X12,1,13,2 X10,11,1,12 X5,15,6,14 X9,19,10,18 X17,3,18,2 X7,16,8,17 X3849 X15,20,16,21 X22,13,11,14 X19,4,20,5 X21,7,22,6
Gauss code {1, 5, -7, 10, -3, 11, -6, 7, -4, -2}, {2, -1, 9, 3, -8, 6, -5, 4, -10, 8, -11, -9}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gif
BraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart3.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart4.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11n244 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) 0 (db)
Jones polynomial -\frac{1}{q^{9/2}}+\frac{1}{q^{7/2}}-q^{5/2}-\frac{1}{q^{5/2}}+2 q^{3/2}-\frac{1}{q^{3/2}}-\frac{1}{q^{13/2}}+\frac{2}{q^{11/2}}-2 \sqrt{q} (db)
Signature -2 (db)
HOMFLY-PT polynomial z^3 a^5+2 z a^5-z^5 a^3-5 z^3 a^3-6 z a^3+z^5 a+5 z^3 a+6 z a+a z^{-1} -z^3 a^{-1} -2 z a^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial -a^3 z^9-a z^9-a^4 z^8-3 a^2 z^8-2 z^8-a^5 z^7+4 a^3 z^7+4 a z^7-z^7 a^{-1} -2 a^6 z^6+5 a^4 z^6+18 a^2 z^6+11 z^6-a^7 z^5+5 a^5 z^5+5 a^3 z^5+4 a z^5+5 z^5 a^{-1} +8 a^6 z^4-2 a^4 z^4-24 a^2 z^4-14 z^4+3 a^7 z^3-5 a^5 z^3-21 a^3 z^3-19 a z^3-6 z^3 a^{-1} -4 a^6 z^2-4 a^4 z^2+4 a^2 z^2+4 z^2+4 a^5 z+12 a^3 z+12 a z+4 z a^{-1} +1-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 \
-6-5-4-3-2-101234χ
6          11
4         1 -1
2        11 0
0      321  2
-2     131   1
-4    222    2
-6   121     0
-8  121      0
-10 111       -1
-12 1         -1
-141          1
Integral Khovanov Homology

(db, data source)

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

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See/edit the Link Page master template (intermediate).

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L11n243.gif

L11n243

L11n245.gif

L11n245