L10a58

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L10a57.gif

L10a57

L10a59.gif

L10a59

Contents

L10a58.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L10a58's Link Presentations]

Planar diagram presentation X8192 X10,3,11,4 X20,16,7,15 X14,5,15,6 X4,13,5,14 X12,17,13,18 X18,11,19,12 X16,20,17,19 X2738 X6,9,1,10
Gauss code {1, -9, 2, -5, 4, -10}, {9, -1, 10, -2, 7, -6, 5, -4, 3, -8, 6, -7, 8, -3}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
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BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart2.gifBraidPart0.gif
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A Morse Link Presentation L10a58 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{u^2 v^3-4 u^2 v^2+2 u^2 v+u v^4-5 u v^3+7 u v^2-5 u v+u+2 v^3-4 v^2+v}{u v^2} (db)
Jones polynomial \frac{11}{q^{9/2}}-\frac{11}{q^{7/2}}+\frac{8}{q^{5/2}}-\frac{6}{q^{3/2}}-\frac{1}{q^{19/2}}+\frac{2}{q^{17/2}}-\frac{5}{q^{15/2}}+\frac{8}{q^{13/2}}-\frac{10}{q^{11/2}}-\sqrt{q}+\frac{3}{\sqrt{q}} (db)
Signature -3 (db)
HOMFLY-PT polynomial a^9 z+a^9 z^{-1} -2 a^7 z^3-3 a^7 z-2 a^7 z^{-1} +a^5 z^5+a^5 z^3+a^5 z+2 a^5 z^{-1} +a^3 z^5+a^3 z^3-a^3 z-a^3 z^{-1} -a z^3-a z (db)
Kauffman polynomial a^{11} z^5-3 a^{11} z^3+2 a^{11} z+2 a^{10} z^6-4 a^{10} z^4+a^{10} z^2+3 a^9 z^7-6 a^9 z^5+5 a^9 z^3-4 a^9 z+a^9 z^{-1} +3 a^8 z^8-6 a^8 z^6+8 a^8 z^4-5 a^8 z^2+a^7 z^9+5 a^7 z^7-17 a^7 z^5+23 a^7 z^3-11 a^7 z+2 a^7 z^{-1} +6 a^6 z^8-13 a^6 z^6+15 a^6 z^4-7 a^6 z^2+a^6+a^5 z^9+6 a^5 z^7-18 a^5 z^5+20 a^5 z^3-9 a^5 z+2 a^5 z^{-1} +3 a^4 z^8-2 a^4 z^6-3 a^4 z^4+a^4 z^2+4 a^3 z^7-7 a^3 z^5+3 a^3 z^3-3 a^3 z+a^3 z^{-1} +3 a^2 z^6-6 a^2 z^4+2 a^2 z^2+a z^5-2 a z^3+a z (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 \
-8-7-6-5-4-3-2-1012χ
2          11
0         2 -2
-2        41 3
-4       53  -2
-6      63   3
-8     55    0
-10    56     -1
-12   35      2
-14  25       -3
-16 14        3
-18 1         -1
-201          1
Integral Khovanov Homology

(db, data source)

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