L11n135

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L11n134.gif

L11n134

L11n136.gif

L11n136

Contents

L11n135.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11n135'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 X20,14,21,13 X22,16,7,15 X4,20,5,19 X14,22,15,21
Gauss code {1, 4, -3, -10, 5, -6}, {6, -1, 7, 3, -2, -5, 8, -11, 9, -7, -4, 2, 10, -8, 11, -9}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart1.gif
BraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gif
A Morse Link Presentation L11n135 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{t(1)^2 t(2)^4-t(1)^2 t(2)^3+t(1) t(2)^3-t(2)^3+t(1)^2 t(2)^2-t(1) t(2)^2+t(2)^2-t(1)^2 t(2)+t(1) t(2)-t(2)+1}{t(1) t(2)^2} (db)
Jones polynomial -3 q^{9/2}+3 q^{7/2}-4 q^{5/2}+3 q^{3/2}-\frac{1}{q^{3/2}}+q^{15/2}-q^{13/2}+2 q^{11/2}-3 \sqrt{q}+\frac{1}{\sqrt{q}} (db)
Signature 1 (db)
HOMFLY-PT polynomial z^5 a^{-5} +5 z^3 a^{-5} +7 z a^{-5} +2 a^{-5} z^{-1} -z^7 a^{-3} -7 z^5 a^{-3} -17 z^3 a^{-3} -16 z a^{-3} -5 a^{-3} z^{-1} +z^5 a^{-1} +5 z^3 a^{-1} +8 z a^{-1} +3 a^{-1} z^{-1} (db)
Kauffman polynomial -z^9 a^{-3} -z^9 a^{-5} -z^8 a^{-2} -2 z^8 a^{-4} -z^8 a^{-6} +7 z^7 a^{-3} +6 z^7 a^{-5} -z^7 a^{-7} +6 z^6 a^{-2} +11 z^6 a^{-4} +4 z^6 a^{-6} -z^6 a^{-8} -2 z^5 a^{-1} -21 z^5 a^{-3} -15 z^5 a^{-5} +4 z^5 a^{-7} -15 z^4 a^{-2} -23 z^4 a^{-4} -4 z^4 a^{-6} +5 z^4 a^{-8} -z^4-a z^3+6 z^3 a^{-1} +29 z^3 a^{-3} +19 z^3 a^{-5} -3 z^3 a^{-7} +13 z^2 a^{-2} +20 z^2 a^{-4} +2 z^2 a^{-6} -6 z^2 a^{-8} +z^2+2 a z-9 z a^{-1} -20 z a^{-3} -9 z a^{-5} -5 a^{-2} -5 a^{-4} + a^{-8} +3 a^{-1} z^{-1} +5 a^{-3} z^{-1} +2 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 \
-2-101234567χ
16         1-1
14          0
12       21 -1
10      1   1
8     22   0
6    21    1
4   12     1
2  22      0
0  2       2
-211        0
-41         1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=0 i=2
r=-2 {\mathbb Z} {\mathbb Z}
r=-1 {\mathbb Z}
r=0 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
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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L11n134

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L11n136