L11n320

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L11n319.gif

L11n319

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L11n321

Contents

L11n320.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11n320's Link Presentations]

Planar diagram presentation X6172 X5,14,6,15 X3849 X15,2,16,3 X16,7,17,8 X9,18,10,19 X4,17,1,18 X19,13,20,22 X13,10,14,11 X21,5,22,12 X11,21,12,20
Gauss code {1, 4, -3, -7}, {-2, -1, 5, 3, -6, 9, -11, 10}, {-9, 2, -4, -5, 7, 6, -8, 11, -10, 8}
A Braid Representative
BraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart3.gif
BraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart0.gif
A Morse Link Presentation L11n320 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(1) t(3)^3 t(2)^3-t(1) t(3)^2 t(2)^3-t(1) t(3)^3 t(2)^2+2 t(1) t(3)^2 t(2)^2-2 t(3) t(2)+t(2)+t(3)-1}{\sqrt{t(1)} t(2)^{3/2} t(3)^{3/2}} (db)
Jones polynomial -q+2-2 q^{-1} +3 q^{-2} -2 q^{-3} +4 q^{-4} -2 q^{-5} +2 q^{-6} - q^{-7} + q^{-8} (db)
Signature -4 (db)
HOMFLY-PT polynomial a^8 z^2+a^8 z^{-2} +2 a^8-a^6 z^6-6 a^6 z^4-11 a^6 z^2-2 a^6 z^{-2} -9 a^6+a^4 z^8+7 a^4 z^6+17 a^4 z^4+19 a^4 z^2+a^4 z^{-2} +9 a^4-a^2 z^6-5 a^2 z^4-6 a^2 z^2-2 a^2 (db)
Kauffman polynomial z^2 a^{10}-a^{10}+z^3 a^9-z a^9+z^4 a^8-2 z^2 a^8-a^8 z^{-2} +3 a^8+z^7 a^7-5 z^5 a^7+9 z^3 a^7-8 z a^7+2 a^7 z^{-1} +2 z^8 a^6-12 z^6 a^6+25 z^4 a^6-25 z^2 a^6-2 a^6 z^{-2} +11 a^6+z^9 a^5-3 z^7 a^5-5 z^5 a^5+17 z^3 a^5-12 z a^5+2 a^5 z^{-1} +4 z^8 a^4-23 z^6 a^4+41 z^4 a^4-32 z^2 a^4-a^4 z^{-2} +11 a^4+z^9 a^3-3 z^7 a^3-5 z^5 a^3+15 z^3 a^3-7 z a^3+2 z^8 a^2-11 z^6 a^2+17 z^4 a^2-10 z^2 a^2+3 a^2+z^7 a-5 z^5 a+6 z^3 a-2 z a (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
3         1-1
1        1 1
-1       11 0
-3     131  1
-5    122   1
-7    42    2
-9  222     2
-11 143      0
-13 12       1
-1511        0
-171         1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-5 i=-3 i=-1
r=-6 {\mathbb Z} {\mathbb Z}
r=-5 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}^{4} {\mathbb Z}^{2}
r=-3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=-2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=0 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=1 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=2 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=3 {\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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L11n319

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L11n321