L10a40

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L10a39.gif

L10a39

L10a41.gif

L10a41

Contents

L10a40.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L10a40's Link Presentations]

Planar diagram presentation X6172 X12,3,13,4 X18,8,19,7 X20,13,5,14 X14,19,15,20 X16,10,17,9 X10,16,11,15 X8,18,9,17 X2536 X4,11,1,12
Gauss code {1, -9, 2, -10}, {9, -1, 3, -8, 6, -7, 10, -2, 4, -5, 7, -6, 8, -3, 5, -4}
A Braid Representative
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A Morse Link Presentation L10a40 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{2 t(2)^3+4 t(1) t(2)^2-7 t(2)^2-7 t(1) t(2)+4 t(2)+2 t(1)}{\sqrt{t(1)} t(2)^{3/2}} (db)
Jones polynomial -\frac{5}{q^{9/2}}-q^{7/2}+\frac{6}{q^{7/2}}+2 q^{5/2}-\frac{8}{q^{5/2}}-4 q^{3/2}+\frac{9}{q^{3/2}}-\frac{1}{q^{13/2}}+\frac{2}{q^{11/2}}+6 \sqrt{q}-\frac{8}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a^7 z^{-1} -3 a^5 z-a^5 z^{-1} +2 a^3 z^3-a^3 z^{-1} -z a^{-3} +3 a z^3+z^3 a^{-1} +3 a z+2 a z^{-1} -z a^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial a^7 z^5-3 a^7 z^3+3 a^7 z-a^7 z^{-1} +2 a^6 z^6-4 a^6 z^4+a^6+2 a^5 z^7-a^5 z^5-6 a^5 z^3+3 a^5 z-a^5 z^{-1} +2 a^4 z^8-3 a^4 z^6+3 a^4 z^4-5 a^4 z^2+2 a^4+a^3 z^9+a^3 z^7-7 a^3 z^5+z^5 a^{-3} +13 a^3 z^3-3 z^3 a^{-3} -8 a^3 z+z a^{-3} +a^3 z^{-1} +5 a^2 z^8-17 a^2 z^6+2 z^6 a^{-2} +26 a^2 z^4-5 z^4 a^{-2} -13 a^2 z^2+z^2 a^{-2} +3 a^2+a z^9+2 a z^7+3 z^7 a^{-1} -15 a z^5-9 z^5 a^{-1} +28 a z^3+9 z^3 a^{-1} -14 a z-5 z a^{-1} +2 a z^{-1} + a^{-1} z^{-1} +3 z^8-10 z^6+14 z^4-7 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 \
-6-5-4-3-2-101234χ
8          11
6         1 -1
4        31 2
2       31  -2
0      53   2
-2     54    -1
-4    34     -1
-6   35      2
-8  23       -1
-10 14        3
-12 1         -1
-141          1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-2 i=0
r=-6 {\mathbb Z}
r=-5 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=-3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=0 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{5}
r=1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=2 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2 {\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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L10a39.gif

L10a39

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L10a41