L11n171

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L11n170.gif

L11n170

L11n172.gif

L11n172

Contents

L11n171.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11n171's Link Presentations]

Planar diagram presentation X8192 X11,21,12,20 X10,4,11,3 X2,17,3,18 X14,5,15,6 X6718 X16,10,17,9 X13,19,14,18 X22,16,7,15 X19,13,20,12 X4,22,5,21
Gauss code {1, -4, 3, -11, 5, -6}, {6, -1, 7, -3, -2, 10, -8, -5, 9, -7, 4, 8, -10, 2, 11, -9}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
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BraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart2.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart0.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11n171 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{t(1)^2 t(2)^4-t(1) t(2)^4-3 t(1)^2 t(2)^3+6 t(1) t(2)^3-t(2)^3+3 t(1)^2 t(2)^2-9 t(1) t(2)^2+3 t(2)^2-t(1)^2 t(2)+6 t(1) t(2)-3 t(2)-t(1)+1}{t(1) t(2)^2} (db)
Jones polynomial 2 q^{11/2}-5 q^{9/2}+9 q^{7/2}-12 q^{5/2}+13 q^{3/2}-14 \sqrt{q}+\frac{10}{\sqrt{q}}-\frac{8}{q^{3/2}}+\frac{4}{q^{5/2}}-\frac{1}{q^{7/2}} (db)
Signature 1 (db)
HOMFLY-PT polynomial z a^{-5} + a^{-5} z^{-1} +z^5 a^{-3} +z^3 a^{-3} -2 z a^{-3} -2 a^{-3} z^{-1} -z^7 a^{-1} +a z^5-4 z^5 a^{-1} +2 a z^3-5 z^3 a^{-1} +a z-z a^{-1} +a z^{-1} (db)
Kauffman polynomial -2 z^9 a^{-1} -2 z^9 a^{-3} -9 z^8 a^{-2} -3 z^8 a^{-4} -6 z^8-7 a z^7-6 z^7 a^{-1} -z^7 a^{-5} -4 a^2 z^6+20 z^6 a^{-2} +6 z^6 a^{-4} +10 z^6-a^3 z^5+14 a z^5+18 z^5 a^{-1} -3 z^5 a^{-5} +6 a^2 z^4-22 z^4 a^{-2} -14 z^4 a^{-4} -3 z^4 a^{-6} -5 z^4+a^3 z^3-7 a z^3-12 z^3 a^{-1} +z^3 a^{-3} +5 z^3 a^{-5} +12 z^2 a^{-2} +15 z^2 a^{-4} +5 z^2 a^{-6} +2 z^2+2 a z-3 z a^{-3} -z a^{-5} -3 a^{-2} -5 a^{-4} -2 a^{-6} +1-a z^{-1} +2 a^{-3} z^{-1} + a^{-5} 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 \
-4-3-2-1012345χ
12         2-2
10        3 3
8       62 -4
6      63  3
4     76   -1
2    76    1
0   48     4
-2  46      -2
-4 15       4
-6 3        -3
-81         1
Integral Khovanov Homology

(db, data source)

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

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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L11n170.gif

L11n170

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L11n172