L10a59

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L10a58.gif

L10a58

L10a60.gif

L10a60

Contents

L10a59.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L10a59's Link Presentations]

Planar diagram presentation X8192 X10,3,11,4 X14,6,15,5 X16,13,17,14 X20,18,7,17 X18,12,19,11 X12,20,13,19 X4,16,5,15 X2738 X6,9,1,10
Gauss code {1, -9, 2, -8, 3, -10}, {9, -1, 10, -2, 6, -7, 4, -3, 8, -4, 5, -6, 7, -5}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gif
BraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart2.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L10a59 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{u^2 v^3-3 u^2 v^2+3 u^2 v-u^2+u v^4-5 u v^3+7 u v^2-5 u v+u-v^4+3 v^3-3 v^2+v}{u v^2} (db)
Jones polynomial q^{11/2}-3 q^{9/2}+6 q^{7/2}-9 q^{5/2}+11 q^{3/2}-12 \sqrt{q}+\frac{10}{\sqrt{q}}-\frac{9}{q^{3/2}}+\frac{5}{q^{5/2}}-\frac{3}{q^{7/2}}+\frac{1}{q^{9/2}} (db)
Signature 1 (db)
HOMFLY-PT polynomial a z^5+z^5 a^{-1} -a^3 z^3+2 a z^3-2 z^3 a^{-3} -a^3 z+3 a z-3 z a^{-1} -z a^{-3} +z a^{-5} +2 a z^{-1} -3 a^{-1} z^{-1} + a^{-3} z^{-1} (db)
Kauffman polynomial -a z^9-z^9 a^{-1} -3 a^2 z^8-3 z^8 a^{-2} -6 z^8-3 a^3 z^7-6 a z^7-8 z^7 a^{-1} -5 z^7 a^{-3} -a^4 z^6+6 a^2 z^6-4 z^6 a^{-2} -5 z^6 a^{-4} +8 z^6+10 a^3 z^5+23 a z^5+20 z^5 a^{-1} +4 z^5 a^{-3} -3 z^5 a^{-5} +3 a^4 z^4+a^2 z^4+15 z^4 a^{-2} +6 z^4 a^{-4} -z^4 a^{-6} +6 z^4-10 a^3 z^3-23 a z^3-17 z^3 a^{-1} -z^3 a^{-3} +3 z^3 a^{-5} -2 a^4 z^2-4 a^2 z^2-12 z^2 a^{-2} -4 z^2 a^{-4} +z^2 a^{-6} -9 z^2+3 a^3 z+11 a z+10 z a^{-1} +z a^{-3} -z a^{-5} +3 a^{-2} + a^{-4} +3-2 a z^{-1} -3 a^{-1} z^{-1} - 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 \
-5-4-3-2-1012345χ
12          1-1
10         2 2
8        41 -3
6       52  3
4      64   -2
2     65    1
0    57     2
-2   45      -1
-4  26       4
-6 13        -2
-8 2         2
-101          -1
Integral Khovanov Homology

(db, data source)

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