L11a117

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L11a116.gif

L11a116

L11a118.gif

L11a118

Contents

L11a117.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11a117's Link Presentations]

Planar diagram presentation X6172 X14,4,15,3 X16,8,17,7 X22,14,5,13 X8,22,9,21 X20,10,21,9 X18,12,19,11 X12,18,13,17 X10,20,11,19 X2536 X4,16,1,15
Gauss code {1, -10, 2, -11}, {10, -1, 3, -5, 6, -9, 7, -8, 4, -2, 11, -3, 8, -7, 9, -6, 5, -4}
A Braid Representative
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A Morse Link Presentation L11a117 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(t(1)-1) (t(2)-1) \left(3 t(2)^2-2 t(2)+3\right)}{\sqrt{t(1)} t(2)^{3/2}} (db)
Jones polynomial q^{21/2}-2 q^{19/2}+4 q^{17/2}-6 q^{15/2}+8 q^{13/2}-10 q^{11/2}+10 q^{9/2}-9 q^{7/2}+6 q^{5/2}-5 q^{3/2}+2 \sqrt{q}-\frac{1}{\sqrt{q}} (db)
Signature 3 (db)
HOMFLY-PT polynomial -z^5 a^{-3} -z^5 a^{-5} -z^5 a^{-7} +z^3 a^{-1} -2 z^3 a^{-3} -z^3 a^{-5} -3 z^3 a^{-7} +z^3 a^{-9} +2 z a^{-1} -z a^{-3} +z a^{-5} -4 z a^{-7} +2 z a^{-9} + a^{-1} z^{-1} - a^{-3} z^{-1} + a^{-5} z^{-1} -2 a^{-7} z^{-1} + a^{-9} z^{-1} (db)
Kauffman polynomial -z^{10} a^{-6} -z^{10} a^{-8} -2 z^9 a^{-5} -5 z^9 a^{-7} -3 z^9 a^{-9} -2 z^8 a^{-4} -z^8 a^{-8} -3 z^8 a^{-10} -2 z^7 a^{-3} +5 z^7 a^{-5} +22 z^7 a^{-7} +13 z^7 a^{-9} -2 z^7 a^{-11} -2 z^6 a^{-2} +2 z^6 a^{-4} +9 z^6 a^{-6} +18 z^6 a^{-8} +12 z^6 a^{-10} -z^6 a^{-12} -z^5 a^{-1} +z^5 a^{-3} -8 z^5 a^{-5} -38 z^5 a^{-7} -21 z^5 a^{-9} +7 z^5 a^{-11} +4 z^4 a^{-2} +z^4 a^{-4} -23 z^4 a^{-6} -39 z^4 a^{-8} -15 z^4 a^{-10} +4 z^4 a^{-12} +3 z^3 a^{-1} +6 z^3 a^{-3} +7 z^3 a^{-5} +25 z^3 a^{-7} +17 z^3 a^{-9} -4 z^3 a^{-11} +15 z^2 a^{-6} +29 z^2 a^{-8} +11 z^2 a^{-10} -3 z^2 a^{-12} -3 z a^{-1} -3 z a^{-3} -5 z a^{-5} -10 z a^{-7} -5 z a^{-9} - a^{-2} -4 a^{-6} -7 a^{-8} -3 a^{-10} + a^{-1} z^{-1} + a^{-3} z^{-1} + a^{-5} z^{-1} +2 a^{-7} z^{-1} + a^{-9} 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 \
-2-10123456789χ
22           1-1
20          1 1
18         31 -2
16        31  2
14       53   -2
12      53    2
10     55     0
8    45      -1
6   25       3
4  34        -1
2 14         3
0 1          -1
-21           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=2 i=4
r=-2 {\mathbb Z}
r=-1 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=0 {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}^{3}
r=1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=4 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=6 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=7 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=8 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=9 {\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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L11a116

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L11a118