L10n46

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L10n45.gif

L10n45

L10n47.gif

L10n47

Contents

L10n46.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L10n46 at Knotilus!


Link Presentations

[edit Notes on L10n46's Link Presentations]

Planar diagram presentation X8192 X9,19,10,18 X6718 X13,20,14,7 X12,5,13,6 X3,10,4,11 X4,15,5,16 X16,12,17,11 X19,14,20,15 X17,2,18,3
Gauss code {1, 10, -6, -7, 5, -3}, {3, -1, -2, 6, 8, -5, -4, 9, 7, -8, -10, 2, -9, 4}
A Braid Representative
BraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart0.gif
BraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gif
A Morse Link Presentation L10n46 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{-u^2 v^2+u v^4-2 u v^3+u v^2-2 u v+u-v^2}{u v^2} (db)
Jones polynomial \frac{4}{q^{9/2}}-\frac{3}{q^{7/2}}+\frac{2}{q^{5/2}}-\frac{2}{q^{3/2}}-\frac{2}{q^{15/2}}+\frac{2}{q^{13/2}}-\frac{3}{q^{11/2}} (db)
Signature -3 (db)
HOMFLY-PT polynomial a^9 z^{-1} -a^7 z^3-3 a^7 z-2 a^7 z^{-1} +a^5 z^5+4 a^5 z^3+5 a^5 z+2 a^5 z^{-1} -2 a^3 z^3-5 a^3 z-a^3 z^{-1} (db)
Kauffman polynomial -3 z^3 a^9+6 z a^9-a^9 z^{-1} -z^6 a^8+2 z^4 a^8-2 z^2 a^8-z^7 a^7+4 z^5 a^7-11 z^3 a^7+10 z a^7-2 a^7 z^{-1} -2 z^6 a^6+4 z^4 a^6-4 z^2 a^6-a^6-z^7 a^5+4 z^5 a^5-11 z^3 a^5+10 z a^5-2 a^5 z^{-1} -z^6 a^4+2 z^4 a^4-2 z^2 a^4-3 z^3 a^3+6 z a^3-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 \
-6-5-4-3-2-10χ
-2      22
-4     110
-6    21 1
-8   21  -1
-10  12   -1
-12 12    1
-1411     0
-162      2
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-4 i=-2
r=-6 {\mathbb Z}^{2} {\mathbb Z}
r=-5 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-2 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-1 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=0 {\mathbb Z}\oplus{\mathbb Z}_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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L10n45

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L10n47