L11n144

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L11n143.gif

L11n143

L11n145.gif

L11n145

Contents

L11n144.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n144 at Knotilus!


Link Presentations

[edit Notes on L11n144's Link Presentations]

Planar diagram presentation X8192 X11,19,12,18 X3,10,4,11 X2,17,3,18 X12,5,13,6 X6718 X16,10,17,9 X20,16,21,15 X22,14,7,13 X14,22,15,21 X19,4,20,5
Gauss code {1, -4, -3, 11, 5, -6}, {6, -1, 7, 3, -2, -5, 9, -10, 8, -7, 4, 2, -11, -8, 10, -9}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart1.gif
BraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11n144 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{2 t(1) t(2)^4-3 t(1) t(2)^3+2 t(2)^3-t(1)^2 t(2)^2+3 t(1) t(2)^2-t(2)^2+2 t(1)^2 t(2)-3 t(1) t(2)+2 t(1)}{t(1) t(2)^2} (db)
Jones polynomial -\frac{4}{q^{9/2}}+\frac{5}{q^{7/2}}+q^{5/2}-\frac{7}{q^{5/2}}-2 q^{3/2}+\frac{6}{q^{3/2}}-\frac{1}{q^{13/2}}+\frac{2}{q^{11/2}}+4 \sqrt{q}-\frac{6}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a^5 z^3+2 a^5 z+a^5 z^{-1} -a^3 z^5-3 a^3 z^3-3 a^3 z-a z^5-3 a z^3+z^3 a^{-1} -4 a z-2 a z^{-1} +2 z a^{-1} + a^{-1} z^{-1} (db)
Kauffman polynomial -a^3 z^9-a z^9-2 a^4 z^8-3 a^2 z^8-z^8-3 a^5 z^7+2 a^3 z^7+5 a z^7-2 a^6 z^6+5 a^4 z^6+12 a^2 z^6+5 z^6-a^7 z^5+10 a^5 z^5-a^3 z^5-14 a z^5-2 z^5 a^{-1} +5 a^6 z^4-3 a^4 z^4-22 a^2 z^4-z^4 a^{-2} -15 z^4+3 a^7 z^3-11 a^5 z^3-4 a^3 z^3+14 a z^3+4 z^3 a^{-1} -a^6 z^2-2 a^4 z^2+13 a^2 z^2+3 z^2 a^{-2} +17 z^2-a^7 z+6 a^5 z+2 a^3 z-7 a z-2 z a^{-1} +a^4-3 a^2-2 a^{-2} -5-a^5 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        1 1
2       31 -2
0      31  2
-2     44   0
-4    32    1
-6   24     2
-8  23      -1
-10  2       2
-1212        -1
-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} {\mathbb Z}
r=-5 {\mathbb Z}^{2}
r=-4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=0 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=1 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=2 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
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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See/edit the Link Page master template (intermediate).

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L11n143.gif

L11n143

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L11n145