L10n104

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L10n103.gif

L10n103

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L10n105

Contents

L10n104.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L10n104 at Knotilus!


Link Presentations

[edit Notes on L10n104's Link Presentations]

Planar diagram presentation X6172 X5,12,6,13 X3849 X15,2,16,3 X7,17,8,16 X14,9,11,10 X20,13,15,14 X19,5,20,10 X11,18,12,19 X4,17,1,18
Gauss code {1, 4, -3, -10}, {-9, 2, 7, -6}, {-2, -1, -5, 3, 6, 8}, {-4, 5, 10, 9, -8, -7}
A Braid Representative
BraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart4.gif
A Morse Link Presentation L10n104 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(t(1) t(2)-t(3) t(4)) (t(3) t(4)-1)}{\sqrt{t(1)} \sqrt{t(2)} t(3) t(4)} (db)
Jones polynomial -\frac{1}{q^{3/2}}-\frac{1}{q^{5/2}}-\frac{2}{q^{7/2}}-\frac{1}{q^{9/2}}-\frac{1}{q^{11/2}}-\frac{1}{q^{13/2}}-\frac{1}{q^{17/2}} (db)
Signature -2 (db)
HOMFLY-PT polynomial a^9 z^{-3} +a^9 z^{-1} -3 a^7 z^{-3} -2 a^7 z-6 a^7 z^{-1} +a^5 z^3+3 a^5 z^{-3} +6 a^5 z+9 a^5 z^{-1} -a^3 z^3-a^3 z^{-3} -4 a^3 z-4 a^3 z^{-1} (db)
Kauffman polynomial a^9 z^7-7 a^9 z^5+15 a^9 z^3-a^9 z^{-3} -12 a^9 z+5 a^9 z^{-1} +a^8 z^6-7 a^8 z^4+14 a^8 z^2+3 a^8 z^{-2} -10 a^8+a^7 z^7-8 a^7 z^5+21 a^7 z^3-3 a^7 z^{-3} -23 a^7 z+12 a^7 z^{-1} +a^6 z^6-8 a^6 z^4+20 a^6 z^2+6 a^6 z^{-2} -19 a^6-a^5 z^5+7 a^5 z^3-3 a^5 z^{-3} -15 a^5 z+12 a^5 z^{-1} -a^4 z^4+6 a^4 z^2+3 a^4 z^{-2} -10 a^4+a^3 z^3-a^3 z^{-3} -4 a^3 z+5 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 \
-8-7-6-5-4-3-2-10χ
-2        11
-4      1 12
-6     14  3
-8      3  3
-10   13    2
-12    2    2
-14  1      1
-161        1
-181        1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-4 i=-2 i=0
r=-8 {\mathbb Z} {\mathbb Z}
r=-7
r=-6 {\mathbb Z}
r=-5 {\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{2} {\mathbb Z}^{3}
r=-3 {\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{3} {\mathbb Z}^{4} {\mathbb Z}
r=-1
r=0 {\mathbb Z} {\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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L10n103.gif

L10n103

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L10n105