L10a99

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L10a98.gif

L10a98

L10a100.gif

L10a100

Contents

L10a99.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L10a99's Link Presentations]

Planar diagram presentation X10,1,11,2 X12,4,13,3 X20,12,9,11 X2,9,3,10 X18,14,19,13 X14,7,15,8 X16,5,17,6 X6,15,7,16 X4,17,5,18 X8,20,1,19
Gauss code {1, -4, 2, -9, 7, -8, 6, -10}, {4, -1, 3, -2, 5, -6, 8, -7, 9, -5, 10, -3}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gif
BraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L10a99 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(2)^2 t(1)^3-2 t(2) t(1)^3+t(1)^3+t(2)^3 t(1)^2-3 t(2)^2 t(1)^2+3 t(2) t(1)^2-2 t(1)^2-2 t(2)^3 t(1)+3 t(2)^2 t(1)-3 t(2) t(1)+t(1)+t(2)^3-2 t(2)^2+t(2)}{t(1)^{3/2} t(2)^{3/2}} (db)
Jones polynomial -q^{9/2}+3 q^{7/2}-5 q^{5/2}+7 q^{3/2}-8 \sqrt{q}+\frac{8}{\sqrt{q}}-\frac{8}{q^{3/2}}+\frac{5}{q^{5/2}}-\frac{4}{q^{7/2}}+\frac{2}{q^{9/2}}-\frac{1}{q^{11/2}} (db)
Signature 1 (db)
HOMFLY-PT polynomial a^5 z+a^5 z^{-1} -2 a^3 z^3-z^3 a^{-3} -4 a^3 z-a^3 z^{-1} -z a^{-3} +a z^5+z^5 a^{-1} +2 a z^3+2 z^3 a^{-1} +a z+z a^{-1} (db)
Kauffman polynomial a^5 z^7-5 a^5 z^5+8 a^5 z^3+z^3 a^{-5} -5 a^5 z+a^5 z^{-1} +2 a^4 z^8-9 a^4 z^6+11 a^4 z^4+3 z^4 a^{-4} -2 a^4 z^2-z^2 a^{-4} -a^4+a^3 z^9-13 a^3 z^5+5 z^5 a^{-3} +21 a^3 z^3-4 z^3 a^{-3} -9 a^3 z+z a^{-3} +a^3 z^{-1} +5 a^2 z^8-17 a^2 z^6+6 z^6 a^{-2} +14 a^2 z^4-8 z^4 a^{-2} -2 a^2 z^2+3 z^2 a^{-2} +a z^9+4 a z^7+5 z^7 a^{-1} -20 a z^5-7 z^5 a^{-1} +19 a z^3+z^3 a^{-1} -6 a z-z a^{-1} +3 z^8-2 z^6-8 z^4+4 z^2 (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
10          11
8         2 -2
6        31 2
4       42  -2
2      43   1
0     55    0
-2    33     0
-4   25      3
-6  23       -1
-8 13        2
-10 1         -1
-121          1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=0 i=2
r=-6 {\mathbb Z}
r=-5 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=-3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=0 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{4}
r=1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
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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L10a98

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L10a100