L11n138

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L11n137.gif

L11n137

L11n139.gif

L11n139

Contents

L11n138.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11n138's Link Presentations]

Planar diagram presentation X8192 X11,19,12,18 X3,10,4,11 X17,3,18,2 X5,13,6,12 X6718 X9,16,10,17 X15,20,16,21 X13,22,14,7 X21,14,22,15 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.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gif
BraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart0.gifBraidPart2.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 L11n138 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(t(2) t(1)-t(1)+1) (t(1) t(2)-t(2)+1)}{t(1) t(2)} (db)
Jones polynomial -\frac{3}{q^{9/2}}+\frac{2}{q^{7/2}}-\frac{3}{q^{5/2}}+\frac{1}{q^{3/2}}-\frac{1}{q^{17/2}}+\frac{2}{q^{15/2}}-\frac{2}{q^{13/2}}+\frac{3}{q^{11/2}}-\frac{1}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a^7 z^3+2 a^7 z-a^5 z^5-4 a^5 z^3-4 a^5 z+a^3 z^3+2 a^3 z+a^3 z^{-1} -a z-a z^{-1} (db)
Kauffman polynomial -z^7 a^9+5 z^5 a^9-6 z^3 a^9+z a^9-2 z^8 a^8+11 z^6 a^8-17 z^4 a^8+8 z^2 a^8-z^9 a^7+4 z^7 a^7-2 z^5 a^7-z^3 a^7-z a^7-3 z^8 a^6+16 z^6 a^6-24 z^4 a^6+12 z^2 a^6-z^9 a^5+5 z^7 a^5-8 z^5 a^5+9 z^3 a^5-5 z a^5-z^8 a^4+5 z^6 a^4-7 z^4 a^4+4 z^2 a^4-z^5 a^3+4 z^3 a^3-4 z a^3+a^3 z^{-1} -a^2-z a+a 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 \
-8-7-6-5-4-3-2-10χ
0        11
-2       110
-4      2  2
-6     12  1
-8    21   1
-10   11    0
-12  12     -1
-14 11      0
-16 1       -1
-181        1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-2 i=0
r=-8 {\mathbb Z}
r=-7 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-6 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-3 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=-1 {\mathbb Z}_2 {\mathbb Z}
r=0 {\mathbb Z} {\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).

See/edit the Link_Splice_Base (expert).

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L11n137.gif

L11n137

L11n139.gif

L11n139