L11a96

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L11a95.gif

L11a95

L11a97.gif

L11a97

Contents

L11a96.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11a96's Link Presentations]

Planar diagram presentation X6172 X12,4,13,3 X20,8,21,7 X22,15,5,16 X16,21,17,22 X18,10,19,9 X14,12,15,11 X10,18,11,17 X8,20,9,19 X2536 X4,14,1,13
Gauss code {1, -10, 2, -11}, {10, -1, 3, -9, 6, -8, 7, -2, 11, -7, 4, -5, 8, -6, 9, -3, 5, -4}
A Braid Representative
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A Morse Link Presentation L11a96 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(t(1)-1) (t(2)-1) \left(2 t(2)^2-7 t(2)+2\right)}{\sqrt{t(1)} t(2)^{3/2}} (db)
Jones polynomial -9 q^{9/2}+12 q^{7/2}-\frac{1}{q^{7/2}}-14 q^{5/2}+\frac{2}{q^{5/2}}+14 q^{3/2}-\frac{6}{q^{3/2}}+q^{15/2}-3 q^{13/2}+5 q^{11/2}-12 \sqrt{q}+\frac{9}{\sqrt{q}} (db)
Signature 1 (db)
HOMFLY-PT polynomial z a^{-7} -2 z^3 a^{-5} -2 z a^{-5} - a^{-5} z^{-1} +z^5 a^{-3} +z^3 a^{-3} +a^3 z+3 z a^{-3} +a^3 z^{-1} +2 a^{-3} z^{-1} +z^5 a^{-1} -2 a z^3-2 a z-z a^{-1} -a z^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial -z^{10} a^{-2} -z^{10} a^{-4} -2 z^9 a^{-1} -5 z^9 a^{-3} -3 z^9 a^{-5} -3 z^8 a^{-2} -4 z^8 a^{-4} -4 z^8 a^{-6} -3 z^8-3 a z^7-2 z^7 a^{-1} +9 z^7 a^{-3} +5 z^7 a^{-5} -3 z^7 a^{-7} -2 a^2 z^6+7 z^6 a^{-2} +16 z^6 a^{-4} +12 z^6 a^{-6} -z^6 a^{-8} +2 z^6-a^3 z^5+3 a z^5+8 z^5 a^{-1} -z^5 a^{-3} +5 z^5 a^{-5} +10 z^5 a^{-7} +3 a^2 z^4-9 z^4 a^{-2} -17 z^4 a^{-4} -9 z^4 a^{-6} +3 z^4 a^{-8} -z^4+3 a^3 z^3+2 a z^3-11 z^3 a^{-1} -15 z^3 a^{-3} -13 z^3 a^{-5} -8 z^3 a^{-7} +7 z^2 a^{-2} +5 z^2 a^{-4} +2 z^2 a^{-6} -z^2 a^{-8} +5 z^2-3 a^3 z-3 a z+8 z a^{-1} +13 z a^{-3} +7 z a^{-5} +2 z a^{-7} -a^2-3 a^{-2} - a^{-4} -2+a^3 z^{-1} +a z^{-1} - a^{-1} 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 \
-4-3-2-101234567χ
16           1-1
14          2 2
12         31 -2
10        62  4
8       63   -3
6      86    2
4     66     0
2    68      -2
0   58       3
-2  14        -3
-4 15         4
-6 1          -1
-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}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=0 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{6}
r=1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=6 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
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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L11a95

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L11a97