L10a64

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L10a63.gif

L10a63

L10a65.gif

L10a65

Contents

L10a64.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L10a64's Link Presentations]

Planar diagram presentation X8192 X10,4,11,3 X20,10,7,9 X2738 X16,14,17,13 X14,5,15,6 X4,15,5,16 X18,12,19,11 X12,18,13,17 X6,20,1,19
Gauss code {1, -4, 2, -7, 6, -10}, {4, -1, 3, -2, 8, -9, 5, -6, 7, -5, 9, -8, 10, -3}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
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BraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gif
A Morse Link Presentation L10a64 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{2 u^2 v^2-4 u^2 v+2 u^2-4 u v^2+7 u v-4 u+2 v^2-4 v+2}{u v} (db)
Jones polynomial -q^{13/2}+3 q^{11/2}-5 q^{9/2}+8 q^{7/2}-10 q^{5/2}+10 q^{3/2}-10 \sqrt{q}+\frac{7}{\sqrt{q}}-\frac{5}{q^{3/2}}+\frac{2}{q^{5/2}}-\frac{1}{q^{7/2}} (db)
Signature 1 (db)
HOMFLY-PT polynomial z^5 a^{-1} +z^5 a^{-3} -2 a z^3+z^3 a^{-1} +2 z^3 a^{-3} -z^3 a^{-5} +a^3 z-3 a z+2 z a^{-3} -z a^{-5} +a^3 z^{-1} -a z^{-1} (db)
Kauffman polynomial -z^9 a^{-1} -z^9 a^{-3} -5 z^8 a^{-2} -3 z^8 a^{-4} -2 z^8-2 a z^7-2 z^7 a^{-1} -4 z^7 a^{-3} -4 z^7 a^{-5} -2 a^2 z^6+10 z^6 a^{-2} +4 z^6 a^{-4} -3 z^6 a^{-6} +z^6-a^3 z^5+3 z^5 a^{-1} +12 z^5 a^{-3} +9 z^5 a^{-5} -z^5 a^{-7} +4 a^2 z^4-11 z^4 a^{-2} +7 z^4 a^{-6} +3 a^3 z^3+7 a z^3-z^3 a^{-1} -13 z^3 a^{-3} -6 z^3 a^{-5} +2 z^3 a^{-7} -a^2 z^2+4 z^2 a^{-2} -2 z^2 a^{-4} -3 z^2 a^{-6} +2 z^2-3 a^3 z-5 a z+4 z a^{-3} +2 z a^{-5} -a^2+a^3 z^{-1} +a 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 \
-4-3-2-10123456χ
14          11
12         2 -2
10        31 2
8       52  -3
6      53   2
4     55    0
2    55     0
0   36      3
-2  24       -2
-4 14        3
-6 1         -1
-81          1
Integral Khovanov Homology

(db, data source)

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

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