L11n191

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L11n190.gif

L11n190

L11n192.gif

L11n192

Contents

L11n191.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n191 at Knotilus!


Link Presentations

[edit Notes on L11n191's Link Presentations]

Planar diagram presentation X8192 X12,3,13,4 X22,10,7,9 X10,14,11,13 X16,6,17,5 X15,21,16,20 X21,19,22,18 X19,15,20,14 X2738 X4,11,5,12 X6,18,1,17
Gauss code {1, -9, 2, -10, 5, -11}, {9, -1, 3, -4, 10, -2, 4, 8, -6, -5, 11, 7, -8, 6, -7, -3}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart1.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart2.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gif
A Morse Link Presentation L11n191 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(1)^2 t(2)^4-2 t(1) t(2)^4+t(2)^4-2 t(1)^2 t(2)^3+2 t(1) t(2)^3-t(2)^3+t(1)^2 t(2)^2-t(1) t(2)^2+t(2)^2-t(1)^2 t(2)+2 t(1) t(2)-2 t(2)+t(1)^2-2 t(1)+1}{t(1) t(2)^2} (db)
Jones polynomial 5 q^{9/2}-6 q^{7/2}+7 q^{5/2}-\frac{1}{q^{5/2}}-7 q^{3/2}+\frac{2}{q^{3/2}}+q^{13/2}-3 q^{11/2}+5 \sqrt{q}-\frac{5}{\sqrt{q}} (db)
Signature 3 (db)
HOMFLY-PT polynomial z a^{-7} -z^5 a^{-5} -4 z^3 a^{-5} -3 z a^{-5} +z^7 a^{-3} +5 z^5 a^{-3} +8 z^3 a^{-3} +6 z a^{-3} + a^{-3} z^{-1} -2 z^5 a^{-1} +a z^3-8 z^3 a^{-1} +3 a z-8 z a^{-1} +2 a z^{-1} -3 a^{-1} z^{-1} (db)
Kauffman polynomial z^2 a^{-8} +3 z^3 a^{-7} -z a^{-7} +z^6 a^{-6} +z^4 a^{-6} -z^2 a^{-6} +3 z^7 a^{-5} -8 z^5 a^{-5} +8 z^3 a^{-5} -3 z a^{-5} +3 z^8 a^{-4} -8 z^6 a^{-4} +4 z^4 a^{-4} - a^{-4} +z^9 a^{-3} +3 z^7 a^{-3} -21 z^5 a^{-3} +23 z^3 a^{-3} -8 z a^{-3} + a^{-3} z^{-1} +5 z^8 a^{-2} -17 z^6 a^{-2} +10 z^4 a^{-2} +4 z^2 a^{-2} -3 a^{-2} +z^9 a^{-1} +a z^7+z^7 a^{-1} -5 a z^5-18 z^5 a^{-1} +9 a z^3+27 z^3 a^{-1} -7 a z-13 z a^{-1} +2 a z^{-1} +3 a^{-1} z^{-1} +2 z^8-8 z^6+7 z^4+2 z^2-3 (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
14         1-1
12        2 2
10       31 -2
8      32  1
6     43   -1
4    33    0
2   35     2
0  22      0
-2 14       3
-4 1        -1
-61         1
Integral Khovanov Homology

(db, data source)

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

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

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L11n190.gif

L11n190

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L11n192