L11n273

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L11n272.gif

L11n272

L11n274.gif

L11n274

Contents

L11n273.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11n273's Link Presentations]

Planar diagram presentation X6172 X10,3,11,4 X16,7,17,8 X8,15,5,16 X11,19,12,18 X17,9,18,22 X21,13,22,12 X13,21,14,20 X19,15,20,14 X2536 X4,9,1,10
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 L11n273 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{2 t(1) t(3)^3+2 t(2) t(3)^3-2 t(3)^3-3 t(1) t(3)^2-3 t(2) t(3)^2+t(3)^2+3 t(1) t(3)-t(1) t(2) t(3)+3 t(2) t(3)-2 t(1)+2 t(1) t(2)-2 t(2)}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{3/2}} (db)
Jones polynomial -2 q^3+3 q^2-6 q+9-8 q^{-1} +9 q^{-2} -6 q^{-3} +6 q^{-4} -2 q^{-5} + q^{-6} (db)
Signature 0 (db)
HOMFLY-PT polynomial a^6 z^{-2} +a^6-3 z^2 a^4-a^4 z^{-2} -4 a^4+2 z^4 a^2+3 z^2 a^2-2 a^2 z^{-2} -a^2+2 z^4+5 z^2+3 z^{-2} +7-2 z^2 a^{-2} - a^{-2} z^{-2} -3 a^{-2} (db)
Kauffman polynomial a^6 z^6-4 a^6 z^4+6 a^6 z^2+a^6 z^{-2} -4 a^6+2 a^5 z^7-5 a^5 z^5+4 a^5 z-2 a^5 z^{-1} +2 a^4 z^8-2 a^4 z^6-8 a^4 z^4+6 a^4 z^2+a^4 z^{-2} -3 a^4+a^3 z^9+3 a^3 z^7-17 a^3 z^5+18 a^3 z^3+3 z^3 a^{-3} -10 a^3 z-7 z a^{-3} +2 a^3 z^{-1} +2 a^{-3} z^{-1} +6 a^2 z^8-19 a^2 z^6+z^6 a^{-2} +19 a^2 z^4-8 a^2 z^2-z^2 a^{-2} -2 a^2 z^{-2} - a^{-2} z^{-2} +4 a^2+ a^{-2} +a z^9+5 a z^7+4 z^7 a^{-1} -29 a z^5-17 z^5 a^{-1} +50 a z^3+35 z^3 a^{-1} -34 a z-27 z a^{-1} +10 a z^{-1} +8 a^{-1} z^{-1} +4 z^8-15 z^6+23 z^4-9 z^2-3 z^{-2} +5 (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 \
-6-5-4-3-2-10123χ
7         2-2
5        1 1
3       52 -3
1      41  3
-1     56   1
-3    441   1
-5   25     3
-7  44      0
-9 15       4
-11 1        -1
-131         1
Integral Khovanov Homology

(db, data source)

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

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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L11n272.gif

L11n272

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L11n274