L10n7

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L10n6.gif

L10n6

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L10n8

Contents

L10n7.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L10n7 at Knotilus!


Link Presentations

[edit Notes on L10n7's Link Presentations]

Planar diagram presentation X6172 X16,7,17,8 X4,17,1,18 X9,14,10,15 X3849 X5,11,6,10 X11,5,12,20 X13,19,14,18 X19,13,20,12 X15,2,16,3
Gauss code {1, 10, -5, -3}, {-6, -1, 2, 5, -4, 6, -7, 9, -8, 4, -10, -2, 3, 8, -9, 7}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gif
BraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L10n7 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{2 u v^3-3 u v^2-3 v+2}{\sqrt{u} v^{3/2}} (db)
Jones polynomial \frac{2}{q^{9/2}}-\frac{3}{q^{7/2}}-q^{5/2}+\frac{3}{q^{5/2}}+q^{3/2}-\frac{3}{q^{3/2}}-\frac{2}{q^{11/2}}-2 \sqrt{q}+\frac{3}{\sqrt{q}} (db)
Signature -3 (db)
HOMFLY-PT polynomial a^7 z^{-1} -z^3 a^5-3 z a^5-a^5 z^{-1} +z^5 a^3+3 z^3 a^3-a^3 z^{-1} +z^5 a+4 z^3 a+4 z a+2 a z^{-1} -z^3 a^{-1} -3 z a^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial 3 a^7 z-a^7 z^{-1} +a^6 z^4+a^6+2 a^5 z^5-4 a^5 z^3+3 a^5 z-a^5 z^{-1} +2 a^4 z^6-4 a^4 z^4-a^4 z^2+2 a^4+2 a^3 z^7-7 a^3 z^5+8 a^3 z^3-7 a^3 z+a^3 z^{-1} +a^2 z^8-3 a^2 z^6+2 a^2 z^4-4 a^2 z^2+3 a^2+3 a z^7+z^7 a^{-1} -15 a z^5-6 z^5 a^{-1} +23 a z^3+11 z^3 a^{-1} -13 a z-6 z a^{-1} +2 a z^{-1} + a^{-1} z^{-1} +z^8-5 z^6+7 z^4-3 z^2+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 \
-4-3-2-101234χ
6        11
4         0
2      21 1
0     1   -1
-2    22   0
-4   22    0
-6  11     0
-8 12      1
-1011       0
-122        2
Integral Khovanov Homology

(db, data source)

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