L11n271

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L11n270.gif

L11n270

L11n272.gif

L11n272

Contents

L11n271.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11n271's Link Presentations]

Planar diagram presentation X6172 X3,11,4,10 X7,17,8,16 X15,5,16,8 X11,19,12,18 X17,9,18,22 X21,13,22,12 X13,21,14,20 X19,15,20,14 X2536 X9,1,10,4
Gauss code {1, -10, -2, 11}, {10, -1, -3, 4}, {-11, 2, -5, 7, -8, 9, -4, 3, -6, 5, -9, 8, -7, 6}
A Braid Representative
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A Morse Link Presentation L11n271 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{-4 u v w^2+5 u v w-2 u v+2 u w^2-2 u w+2 v w^2-2 v w+2 w^3-5 w^2+4 w}{\sqrt{u} \sqrt{v} w^{3/2}} (db)
Jones polynomial -q^{10}+2 q^9-5 q^8+7 q^7-9 q^6+11 q^5-8 q^4+9 q^3-5 q^2+3 q (db)
Signature 2 (db)
HOMFLY-PT polynomial - a^{-10} z^{-2} - a^{-10} +3 z^2 a^{-8} +3 a^{-8} z^{-2} +5 a^{-8} -2 z^4 a^{-6} -4 z^2 a^{-6} -2 a^{-6} z^{-2} -5 a^{-6} -2 z^4 a^{-4} -3 z^2 a^{-4} - a^{-4} z^{-2} -2 a^{-4} +3 z^2 a^{-2} + a^{-2} z^{-2} +3 a^{-2} (db)
Kauffman polynomial z^7 a^{-11} -5 z^5 a^{-11} +9 z^3 a^{-11} -7 z a^{-11} +2 a^{-11} z^{-1} +2 z^8 a^{-10} -8 z^6 a^{-10} +9 z^4 a^{-10} -2 z^2 a^{-10} - a^{-10} z^{-2} + a^{-10} +z^9 a^{-9} +3 z^7 a^{-9} -26 z^5 a^{-9} +43 z^3 a^{-9} -27 z a^{-9} +8 a^{-9} z^{-1} +7 z^8 a^{-8} -23 z^6 a^{-8} +20 z^4 a^{-8} -8 z^2 a^{-8} -3 a^{-8} z^{-2} +5 a^{-8} +z^9 a^{-7} +10 z^7 a^{-7} -43 z^5 a^{-7} +52 z^3 a^{-7} -34 z a^{-7} +10 a^{-7} z^{-1} +5 z^8 a^{-6} -8 z^6 a^{-6} -z^4 a^{-6} -3 z^2 a^{-6} -2 a^{-6} z^{-2} +4 a^{-6} +8 z^7 a^{-5} -19 z^5 a^{-5} +18 z^3 a^{-5} -10 z a^{-5} +2 a^{-5} z^{-1} +7 z^6 a^{-4} -12 z^4 a^{-4} +9 z^2 a^{-4} + a^{-4} z^{-2} -3 a^{-4} +3 z^5 a^{-3} +4 z a^{-3} -2 a^{-3} z^{-1} +6 z^2 a^{-2} + a^{-2} z^{-2} -4 a^{-2} (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 \
0123456789χ
21         1-1
19        1 1
17       41 -3
15      31  2
13     64   -2
11    53    2
9   47     3
7  54      1
5  4       4
335        -2
13         3
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=1 i=3
r=0 {\mathbb Z}^{3} {\mathbb Z}^{3}
r=1 {\mathbb Z}^{5}
r=2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=4 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{5}
r=5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=6 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=7 {\mathbb Z}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
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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L11n270.gif

L11n270

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L11n272