L11n179

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L11n178.gif

L11n178

L11n180.gif

L11n180

Contents

L11n179.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n179 at Knotilus!


Link Presentations

[edit Notes on L11n179's Link Presentations]

Planar diagram presentation X8192 X3,10,4,11 X16,7,17,8 X20,10,21,9 X22,15,7,16 X14,6,15,5 X18,14,19,13 X12,20,13,19 X11,4,12,5 X6,18,1,17 X2,21,3,22
Gauss code {1, -11, -2, 9, 6, -10}, {3, -1, 4, 2, -9, -8, 7, -6, 5, -3, 10, -7, 8, -4, 11, -5}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart1.gif
BraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart4.gifBraidPart3.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11n179 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{t(1) t(2)^4-t(2)^4+t(1)^2 t(2)^3-4 t(1) t(2)^3+2 t(2)^3-2 t(1)^2 t(2)^2+7 t(1) t(2)^2-2 t(2)^2+2 t(1)^2 t(2)-4 t(1) t(2)+t(2)-t(1)^2+t(1)}{t(1) t(2)^2} (db)
Jones polynomial -\frac{6}{q^{9/2}}+\frac{8}{q^{7/2}}+q^{5/2}-\frac{10}{q^{5/2}}-4 q^{3/2}+\frac{10}{q^{3/2}}-\frac{1}{q^{13/2}}+\frac{3}{q^{11/2}}+6 \sqrt{q}-\frac{9}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial z^3 a^5+z a^5+a^5 z^{-1} -z^5 a^3-2 z^3 a^3-3 z a^3-2 a^3 z^{-1} -z^5 a-z^3 a+z a+2 a z^{-1} +z^3 a^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial -a^3 z^9-a z^9-4 a^4 z^8-5 a^2 z^8-z^8-5 a^5 z^7-5 a^3 z^7-3 a^6 z^6+9 a^4 z^6+12 a^2 z^6-a^7 z^5+13 a^5 z^5+18 a^3 z^5-4 z^5 a^{-1} +6 a^6 z^4-8 a^4 z^4-15 a^2 z^4-z^4 a^{-2} -2 z^4+2 a^7 z^3-11 a^5 z^3-23 a^3 z^3-5 a z^3+5 z^3 a^{-1} -a^6 z^2+3 a^4 z^2+6 a^2 z^2+z^2 a^{-2} +3 z^2+5 a^5 z+11 a^3 z+7 a z+z a^{-1} -a^2-a^5 z^{-1} -2 a^3 z^{-1} -2 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 \
-6-5-4-3-2-10123χ
6         1-1
4        3 3
2       31 -2
0      63  3
-2     54   -1
-4    55    0
-6   46     2
-8  24      -2
-10 14       3
-12 2        -2
-141         1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-2 i=0
r=-6 {\mathbb Z}
r=-5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{5}
r=-1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=0 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{6}
r=1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=2 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=3 {\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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L11n178

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L11n180