L11n458

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L11n457.gif

L11n457

L11n459.gif

L11n459

Contents

L11n458.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n458 at Knotilus!


Link L11n458.
A graph, L11n458.
A part of a knot and a part of a graph.

Link Presentations

[edit Notes on L11n458's Link Presentations]

Planar diagram presentation X6172 X3,13,4,12 X7,21,8,20 X19,5,20,10 X13,19,14,22 X21,11,22,18 X17,15,18,14 X9,17,10,16 X15,9,16,8 X2536 X11,1,12,4
Gauss code {1, -10, -2, 11}, {-4, 3, -6, 5}, {10, -1, -3, 9, -8, 4}, {-11, 2, -5, 7, -9, 8, -7, 6}
A Braid Representative
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A Morse Link Presentation L11n458 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{t(1) t(4)^2 t(3)^2-t(4)^2 t(3)^2+t(1) t(3)^2-t(1) t(2) t(3)^2-2 t(1) t(4) t(3)^2+2 t(1) t(2) t(4) t(3)^2+t(4) t(3)^2-t(1) t(4)^2 t(3)-2 t(2) t(4)^2 t(3)+2 t(4)^2 t(3)-2 t(1) t(3)+2 t(1) t(2) t(3)-t(2) t(3)+3 t(1) t(4) t(3)-3 t(1) t(2) t(4) t(3)+3 t(2) t(4) t(3)-3 t(4) t(3)+t(2) t(4)^2-t(4)^2-t(1) t(2)+t(2)+t(1) t(2) t(4)-2 t(2) t(4)+2 t(4)}{\sqrt{t(1)} \sqrt{t(2)} t(3) t(4)} (db)
Jones polynomial 11 q^{9/2}-11 q^{7/2}+5 q^{5/2}-3 q^{3/2}+q^{21/2}-4 q^{19/2}+7 q^{17/2}-11 q^{15/2}+12 q^{13/2}-15 q^{11/2} (db)
Signature 3 (db)
HOMFLY-PT polynomial z^3 a^{-9} - a^{-9} z^{-3} - a^{-9} z^{-1} -z^5 a^{-7} +3 a^{-7} z^{-3} +4 z a^{-7} +6 a^{-7} z^{-1} -2 z^5 a^{-5} -6 z^3 a^{-5} -3 a^{-5} z^{-3} -10 z a^{-5} -9 a^{-5} z^{-1} +3 z^3 a^{-3} + a^{-3} z^{-3} +6 z a^{-3} +4 a^{-3} z^{-1} (db)
Kauffman polynomial z^6 a^{-12} -2 z^4 a^{-12} +z^2 a^{-12} +4 z^7 a^{-11} -11 z^5 a^{-11} +8 z^3 a^{-11} -2 z a^{-11} +5 z^8 a^{-10} -11 z^6 a^{-10} +2 z^4 a^{-10} +2 z^2 a^{-10} +2 z^9 a^{-9} +7 z^7 a^{-9} -33 z^5 a^{-9} +32 z^3 a^{-9} - a^{-9} z^{-3} -16 z a^{-9} +5 a^{-9} z^{-1} +11 z^8 a^{-8} -25 z^6 a^{-8} +9 z^4 a^{-8} +10 z^2 a^{-8} +3 a^{-8} z^{-2} -10 a^{-8} +2 z^9 a^{-7} +10 z^7 a^{-7} -41 z^5 a^{-7} +52 z^3 a^{-7} -3 a^{-7} z^{-3} -31 z a^{-7} +12 a^{-7} z^{-1} +6 z^8 a^{-6} -10 z^6 a^{-6} +2 z^4 a^{-6} +18 z^2 a^{-6} +6 a^{-6} z^{-2} -19 a^{-6} +7 z^7 a^{-5} -19 z^5 a^{-5} +34 z^3 a^{-5} -3 a^{-5} z^{-3} -27 z a^{-5} +12 a^{-5} z^{-1} +3 z^6 a^{-4} -3 z^4 a^{-4} +9 z^2 a^{-4} +3 a^{-4} z^{-2} -10 a^{-4} +6 z^3 a^{-3} - a^{-3} z^{-3} -10 z a^{-3} +5 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 \
0123456789χ
22         1-1
20        3 3
18       41 -3
16      73  4
14     76   -1
12    85    3
10   59     4
8  66      0
6 17       6
424        -2
23         3
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=2 i=4
r=0 {\mathbb Z}^{3} {\mathbb Z}^{2}
r=1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}
r=2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{6}
r=3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=4 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{8}
r=5 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=6 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{7}
r=7 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
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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L11n457

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L11n459