L10n45

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L10n44.gif

L10n44

L10n46.gif

L10n46

Contents

L10n45.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L10n45 at Knotilus!


Link Presentations

[edit Notes on L10n45's Link Presentations]

Planar diagram presentation X8192 X18,9,19,10 X6718 X20,14,7,13 X5,13,6,12 X3,10,4,11 X15,5,16,4 X11,16,12,17 X14,20,15,19 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
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gif
A Morse Link Presentation L10n45 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(2)^2 t(1)^2-t(2) t(1)^2-t(2) t(1)-t(2)+1}{t(1) t(2)} (db)
Jones polynomial -q^{5/2}+q^{3/2}-\sqrt{q}+\frac{1}{\sqrt{q}}-\frac{1}{q^{3/2}}-\frac{1}{q^{7/2}} (db)
Signature -3 (db)
HOMFLY-PT polynomial a^5 z^{-1} -a^3 z^3-4 a^3 z-2 a^3 z^{-1} +a z^5+5 a z^3-z^3 a^{-1} +6 a z+2 a z^{-1} -3 z a^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial -a^5 z+a^5 z^{-1} -a^4 z^2+a^3 z^7-6 a^3 z^5+10 a^3 z^3-7 a^3 z+2 a^3 z^{-1} +a^2 z^8-6 a^2 z^6+10 a^2 z^4-6 a^2 z^2+a^2+2 a z^7+z^7 a^{-1} -12 a z^5-6 z^5 a^{-1} +20 a z^3+10 z^3 a^{-1} -11 a z-5 z a^{-1} +2 a z^{-1} + a^{-1} z^{-1} +z^8-6 z^6+10 z^4-5 z^2 (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 \
-2-101234χ
6      11
4       0
2    11 0
0  11   0
-2  11   0
-4111    1
-61      1
-81      1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-4 i=-2 i=0
r=-2 {\mathbb Z} {\mathbb Z} {\mathbb Z}
r=-1 {\mathbb Z}_2 {\mathbb Z}
r=0 {\mathbb Z} {\mathbb Z} {\mathbb Z}
r=1 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=2 {\mathbb Z}_2 {\mathbb Z}
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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L10n44

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L10n46