L11n143

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L11n142.gif

L11n142

L11n144.gif

L11n144

Contents

L11n143.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11n143's Link Presentations]

Planar diagram presentation X8192 X11,19,12,18 X3,10,4,11 X17,3,18,2 X12,5,13,6 X6718 X16,10,17,9 X20,16,21,15 X22,14,7,13 X14,22,15,21 X4,20,5,19
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.gifBraidPart1.gifBraidPart0.gifBraidPart1.gif
BraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart2.gifBraidPart3.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.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 L11n143 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{2 u^2 v^2-2 u^2 v+u v-2 v+2}{u v} (db)
Jones polynomial q^{13/2}-q^{11/2}+2 q^{9/2}-3 q^{7/2}+2 q^{5/2}-3 q^{3/2}+2 \sqrt{q}-\frac{2}{\sqrt{q}}+\frac{1}{q^{3/2}}-\frac{1}{q^{5/2}} (db)
Signature -1 (db)
HOMFLY-PT polynomial z^3 a^{-5} +3 z a^{-5} + a^{-5} z^{-1} -z^5 a^{-3} -4 z^3 a^{-3} -4 z a^{-3} -2 a^{-3} z^{-1} -z^5 a^{-1} +a z^3-4 z^3 a^{-1} +3 a z-3 z a^{-1} +a z^{-1} (db)
Kauffman polynomial z^8 a^{-6} -7 z^6 a^{-6} +16 z^4 a^{-6} -13 z^2 a^{-6} +2 a^{-6} +z^9 a^{-5} -6 z^7 a^{-5} +11 z^5 a^{-5} -8 z^3 a^{-5} +4 z a^{-5} - a^{-5} z^{-1} +3 z^8 a^{-4} -18 z^6 a^{-4} +33 z^4 a^{-4} -22 z^2 a^{-4} +5 a^{-4} +z^9 a^{-3} -4 z^7 a^{-3} +2 z^5 a^{-3} +a^3 z+5 z a^{-3} -2 a^{-3} z^{-1} +2 z^8 a^{-2} -10 z^6 a^{-2} +14 z^4 a^{-2} +a^2 z^2-9 z^2 a^{-2} +3 a^{-2} +2 z^7 a^{-1} -9 z^5 a^{-1} +2 a z^3+10 z^3 a^{-1} -4 a z-4 z a^{-1} +a z^{-1} +z^6-3 z^4+z^2-1 (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 \
-2-101234567χ
14         1-1
12          0
10       21 -1
8      1   1
6     12   1
4    21    1
2    1     1
0  22      0
-2  1       1
-411        0
-61         1
Integral Khovanov Homology

(db, data source)

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

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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L11n142.gif

L11n142

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L11n144