L11a233

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L11a232

L11a234.gif

L11a234

Contents

L11a233.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11a233's Link Presentations]

Planar diagram presentation X8192 X12,4,13,3 X18,6,19,5 X20,12,21,11 X22,16,7,15 X10,18,11,17 X16,22,17,21 X14,10,15,9 X4,20,5,19 X2738 X6,14,1,13
Gauss code {1, -10, 2, -9, 3, -11}, {10, -1, 8, -6, 4, -2, 11, -8, 5, -7, 6, -3, 9, -4, 7, -5}
A Braid Representative
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A Morse Link Presentation L11a233 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{3 u^2 v^3-7 u^2 v^2+6 u^2 v-2 u^2+2 u v^4-11 u v^3+17 u v^2-11 u v+2 u-2 v^4+6 v^3-7 v^2+3 v}{u v^2} (db)
Jones polynomial 25 q^{9/2}-23 q^{7/2}+16 q^{5/2}-10 q^{3/2}+q^{21/2}-4 q^{19/2}+10 q^{17/2}-16 q^{15/2}+22 q^{13/2}-26 q^{11/2}+4 \sqrt{q}-\frac{1}{\sqrt{q}} (db)
Signature 3 (db)
HOMFLY-PT polynomial z^3 a^{-9} + a^{-9} z^{-1} -z^5 a^{-7} +z^3 a^{-7} -2 a^{-7} z^{-1} -3 z^5 a^{-5} -5 z^3 a^{-5} -3 z a^{-5} -z^5 a^{-3} +2 z^3 a^{-3} +4 z a^{-3} + a^{-3} z^{-1} +z^3 a^{-1} (db)
Kauffman polynomial z^6 a^{-12} -2 z^4 a^{-12} +z^2 a^{-12} +4 z^7 a^{-11} -8 z^5 a^{-11} +4 z^3 a^{-11} +8 z^8 a^{-10} -18 z^6 a^{-10} +14 z^4 a^{-10} -7 z^2 a^{-10} +2 a^{-10} +8 z^9 a^{-9} -13 z^7 a^{-9} +5 z^5 a^{-9} -4 z^3 a^{-9} +2 z a^{-9} - a^{-9} z^{-1} +3 z^{10} a^{-8} +13 z^8 a^{-8} -43 z^6 a^{-8} +39 z^4 a^{-8} -19 z^2 a^{-8} +5 a^{-8} +17 z^9 a^{-7} -31 z^7 a^{-7} +14 z^5 a^{-7} -z^3 a^{-7} +2 z a^{-7} -2 a^{-7} z^{-1} +3 z^{10} a^{-6} +17 z^8 a^{-6} -48 z^6 a^{-6} +41 z^4 a^{-6} -14 z^2 a^{-6} +3 a^{-6} +9 z^9 a^{-5} -5 z^7 a^{-5} -14 z^5 a^{-5} +18 z^3 a^{-5} -5 z a^{-5} +12 z^8 a^{-4} -20 z^6 a^{-4} +14 z^4 a^{-4} -3 z^2 a^{-4} - a^{-4} +9 z^7 a^{-3} -14 z^5 a^{-3} +10 z^3 a^{-3} -5 z a^{-3} + a^{-3} z^{-1} +4 z^6 a^{-2} -4 z^4 a^{-2} +z^5 a^{-1} -z^3 a^{-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          3 3
18         71 -6
16        93  6
14       137   -6
12      139    4
10     1213     1
8    1113      -2
6   613       7
4  410        -6
2 17         6
0 3          -3
-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}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=0 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{4}
r=1 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=2 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{11}
r=3 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=4 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{13}
r=5 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{13}
r=6 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=7 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=8 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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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L11a232

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L11a234