L10n57

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L10n56

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L10n58

Contents

L10n57.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L10n57's Link Presentations]

Planar diagram presentation X10,1,11,2 X20,5,9,6 X3,15,4,14 X15,5,16,4 X7,17,8,16 X11,18,12,19 X17,12,18,13 X2,9,3,10 X13,1,14,8 X6,19,7,20
Gauss code {1, -8, -3, 4, 2, -10, -5, 9}, {8, -1, -6, 7, -9, 3, -4, 5, -7, 6, 10, -2}
A Braid Representative
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A Morse Link Presentation L10n57 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(u-1) (u+1)^2 (v-1)}{u^{3/2} \sqrt{v}} (db)
Jones polynomial \frac{1}{q^{9/2}}-q^{7/2}-\frac{1}{q^{7/2}}+q^{5/2}-\frac{1}{q^{3/2}}-\frac{1}{\sqrt{q}} (db)
Signature 0 (db)
HOMFLY-PT polynomial -a^3 z^3-3 a^3 z-z a^{-3} +a z^5+5 a z^3+5 a z-z a^{-1} +a z^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial -a^3 z^7-a z^7-a^4 z^6-a^2 z^6-z^6 a^{-2} -z^6+6 a^3 z^5+7 a z^5-z^5 a^{-3} +5 a^4 z^4+6 a^2 z^4+5 z^4 a^{-2} +6 z^4-10 a^3 z^3-14 a z^3+4 z^3 a^{-3} -5 a^4 z^2-7 a^2 z^2-5 z^2 a^{-2} -7 z^2+6 a^3 z+10 a z+2 z a^{-1} -2 z a^{-3} +1-a z^{-1} - a^{-1} 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 \
-5-4-3-2-101234χ
8         11
6          0
4      111 -1
2     11   0
0    131   1
-2   112    2
-4   1      1
-6 111      1
-8          0
-101         -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-2 i=0 i=2
r=-5 {\mathbb Z}
r=-4 {\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}
r=-2 {\mathbb Z} {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}_2 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=0 {\mathbb Z}^{2} {\mathbb Z}^{3} {\mathbb Z}
r=1 {\mathbb Z}\oplus{\mathbb Z}_2 {\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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L10n56.gif

L10n56

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L10n58