L10n30

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L10n29.gif

L10n29

L10n31.gif

L10n31

Contents

L10n30.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L10n30's Link Presentations]

Planar diagram presentation X6172 X3,10,4,11 X15,5,16,20 X7,17,8,16 X17,12,18,13 X9,14,10,15 X13,18,14,19 X19,9,20,8 X2536 X11,4,12,1
Gauss code {1, -9, -2, 10}, {9, -1, -4, 8, -6, 2, -10, 5, -7, 6, -3, 4, -5, 7, -8, 3}
A Braid Representative
BraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gif
BraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart4.gif
BraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L10n30 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{t(1) t(2)^5-3 t(1) t(2)^4+t(2)^4+4 t(1) t(2)^3-2 t(2)^3-2 t(1) t(2)^2+4 t(2)^2+t(1) t(2)-3 t(2)+1}{\sqrt{t(1)} t(2)^{5/2}} (db)
Jones polynomial \frac{6}{q^{9/2}}-\frac{8}{q^{7/2}}+\frac{7}{q^{5/2}}+q^{3/2}-\frac{7}{q^{3/2}}+\frac{2}{q^{13/2}}-\frac{5}{q^{11/2}}-3 \sqrt{q}+\frac{5}{\sqrt{q}} (db)
Signature -3 (db)
HOMFLY-PT polynomial -z a^7-a^7 z^{-1} +z^5 a^5+4 z^3 a^5+6 z a^5+4 a^5 z^{-1} -z^7 a^3-5 z^5 a^3-9 z^3 a^3-9 z a^3-4 a^3 z^{-1} +z^5 a+3 z^3 a+2 z a+a z^{-1} (db)
Kauffman polynomial -3 z^2 a^8+a^8-z^5 a^7-5 z^3 a^7+4 z a^7-a^7 z^{-1} -5 z^6 a^6+11 z^4 a^6-13 z^2 a^6+4 a^6-6 z^7 a^5+18 z^5 a^5-24 z^3 a^5+16 z a^5-4 a^5 z^{-1} -2 z^8 a^4-3 z^6 a^4+21 z^4 a^4-21 z^2 a^4+7 a^4-9 z^7 a^3+29 z^5 a^3-28 z^3 a^3+15 z a^3-4 a^3 z^{-1} -2 z^8 a^2+z^6 a^2+13 z^4 a^2-14 z^2 a^2+4 a^2-3 z^7 a+10 z^5 a-9 z^3 a+3 z a-a z^{-1} -z^6+3 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 \
-5-4-3-2-10123χ
4        1-1
2       2 2
0      31 -2
-2     42  2
-4    44   0
-6   43    1
-8  24     2
-10 34      -1
-12 3       3
-142        -2
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-4 i=-2
r=-5 {\mathbb Z}^{2}
r=-4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=-3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=0 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{4}
r=1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=2 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
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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L10n29.gif

L10n29

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L10n31