L11n459

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L11n458.gif

L11n458

L11n459.gif

L11n459

Contents

L11n459.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n459 at Knotilus!


Link L11n459.
A graph, L11n459.
A part of a knot and a part of a graph.

Link Presentations

[edit Notes on L11n459's Link Presentations]

Planar diagram presentation X6172 X12,3,13,4 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 X4,11,1,12
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 L11n459 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(w x-1) (u w x+u (-w)-u x+2 u-2 v w x+v w+v x-v)}{\sqrt{u} \sqrt{v} w x} (db)
Jones polynomial -2 q^{9/2}-q^{5/2}-\frac{1}{q^{5/2}}-2 q^{3/2}+\frac{1}{q^{3/2}}+q^{15/2}-2 q^{13/2}+q^{11/2}-\frac{3}{\sqrt{q}} (db)
Signature 2 (db)
HOMFLY-PT polynomial z a^{-7} -z^3 a^{-5} - a^{-5} z^{-3} -2 z a^{-5} -2 a^{-5} z^{-1} +z^3 a^{-3} +3 a^{-3} z^{-3} +6 z a^{-3} +7 a^{-3} z^{-1} -z^5 a^{-1} +a z^3-5 z^3 a^{-1} +a z^{-3} -3 a^{-1} z^{-3} +3 a z-8 z a^{-1} +3 a z^{-1} -8 a^{-1} z^{-1} (db)
Kauffman polynomial -z^9 a^{-1} -z^9 a^{-3} -3 z^8 a^{-2} -3 z^8 a^{-4} -z^8 a^{-6} -z^8-a z^7+4 z^7 a^{-1} +3 z^7 a^{-3} -4 z^7 a^{-5} -2 z^7 a^{-7} +17 z^6 a^{-2} +18 z^6 a^{-4} +4 z^6 a^{-6} -z^6 a^{-8} +4 z^6+6 a z^5+3 z^5 a^{-1} +12 z^5 a^{-3} +25 z^5 a^{-5} +10 z^5 a^{-7} -18 z^4 a^{-2} -23 z^4 a^{-4} +4 z^4 a^{-8} +z^4-11 a z^3-19 z^3 a^{-1} -35 z^3 a^{-3} -39 z^3 a^{-5} -12 z^3 a^{-7} -10 z^2 a^{-2} -2 z^2 a^{-6} -2 z^2 a^{-8} -10 z^2+10 a z+23 z a^{-1} +27 z a^{-3} +20 z a^{-5} +6 z a^{-7} +19 a^{-2} +10 a^{-4} +10-5 a z^{-1} -12 a^{-1} z^{-1} -12 a^{-3} z^{-1} -5 a^{-5} z^{-1} -6 a^{-2} z^{-2} -3 a^{-4} z^{-2} -3 z^{-2} +a z^{-3} +3 a^{-1} z^{-3} +3 a^{-3} z^{-3} + a^{-5} z^{-3} (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 \
-4-3-2-101234567χ
16           1-1
14          1 1
12        111 1
10       131  1
8      222   2
6     362    1
4    124     3
2   251      2
0  113       3
-2  2         2
-411          0
-61           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=0 i=2 i=4
r=-4 {\mathbb Z} {\mathbb Z}
r=-3 {\mathbb Z}
r=-2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=0 {\mathbb Z}^{3} {\mathbb Z}^{5}\oplus{\mathbb Z}_2 {\mathbb Z}
r=1 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=6 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=7 {\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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L11n458.gif

L11n458

L11n459.gif

L11n459