L11a242

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L11a241.gif

L11a241

L11a243.gif

L11a243

Contents

L11a242.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11a242's Link Presentations]

Planar diagram presentation X8192 X12,4,13,3 X22,10,7,9 X14,12,15,11 X10,16,11,15 X18,6,19,5 X20,18,21,17 X16,22,17,21 X2738 X4,14,5,13 X6,20,1,19
Gauss code {1, -9, 2, -10, 6, -11}, {9, -1, 3, -5, 4, -2, 10, -4, 5, -8, 7, -6, 11, -7, 8, -3}
A Braid Representative
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A Morse Link Presentation L11a242 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{4 t(2)^2 t(1)^2-7 t(2) t(1)^2+3 t(1)^2-7 t(2)^2 t(1)+13 t(2) t(1)-7 t(1)+3 t(2)^2-7 t(2)+4}{t(1) t(2)} (db)
Jones polynomial q^{21/2}-4 q^{19/2}+8 q^{17/2}-12 q^{15/2}+16 q^{13/2}-18 q^{11/2}+17 q^{9/2}-15 q^{7/2}+10 q^{5/2}-6 q^{3/2}+2 \sqrt{q}-\frac{1}{\sqrt{q}} (db)
Signature 3 (db)
HOMFLY-PT polynomial z^3 a^{-9} -z^5 a^{-7} +z a^{-7} -2 z^5 a^{-5} -3 z^3 a^{-5} -z a^{-5} -z^5 a^{-3} -z^3 a^{-3} -z a^{-3} - a^{-3} z^{-1} +z^3 a^{-1} +2 z a^{-1} + a^{-1} z^{-1} (db)
Kauffman polynomial -z^{10} a^{-6} -z^{10} a^{-8} -3 z^9 a^{-5} -7 z^9 a^{-7} -4 z^9 a^{-9} -4 z^8 a^{-4} -9 z^8 a^{-6} -11 z^8 a^{-8} -6 z^8 a^{-10} -3 z^7 a^{-3} -z^7 a^{-5} +7 z^7 a^{-7} +z^7 a^{-9} -4 z^7 a^{-11} -2 z^6 a^{-2} +5 z^6 a^{-4} +23 z^6 a^{-6} +31 z^6 a^{-8} +14 z^6 a^{-10} -z^6 a^{-12} -z^5 a^{-1} +2 z^5 a^{-3} +7 z^5 a^{-5} +10 z^5 a^{-7} +16 z^5 a^{-9} +10 z^5 a^{-11} +3 z^4 a^{-2} -6 z^4 a^{-4} -23 z^4 a^{-6} -23 z^4 a^{-8} -7 z^4 a^{-10} +2 z^4 a^{-12} +3 z^3 a^{-1} +3 z^3 a^{-3} -9 z^3 a^{-5} -16 z^3 a^{-7} -13 z^3 a^{-9} -6 z^3 a^{-11} +4 z^2 a^{-4} +7 z^2 a^{-6} +5 z^2 a^{-8} +z^2 a^{-10} -z^2 a^{-12} -3 z a^{-1} -3 z a^{-3} +3 z a^{-5} +5 z a^{-7} +3 z a^{-9} +z a^{-11} - a^{-2} + a^{-1} z^{-1} + a^{-3} 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          3 3
18         51 -4
16        73  4
14       95   -4
12      97    2
10     89     1
8    79      -2
6   49       5
4  26        -4
2 15         4
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}^{5}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=2 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{7}
r=3 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=4 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=5 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=6 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=7 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
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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L11a241

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L11a243