L10a69

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L10a68.gif

L10a68

L10a70.gif

L10a70

Contents

L10a69.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L10a69 at Knotilus!


Link Presentations

[edit Notes on L10a69's Link Presentations]

Planar diagram presentation X8192 X16,9,17,10 X6718 X20,17,7,18 X10,4,11,3 X14,6,15,5 X4,12,5,11 X18,14,19,13 X12,20,13,19 X2,16,3,15
Gauss code {1, -10, 5, -7, 6, -3}, {3, -1, 2, -5, 7, -9, 8, -6, 10, -2, 4, -8, 9, -4}
A Braid Representative
BraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gif
BraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gif
BraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L10a69 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{\left(v^2-3 v+1\right) (u v-u+1) (u v-v+1)}{u v^2} (db)
Jones polynomial -q^{13/2}+3 q^{11/2}-7 q^{9/2}+11 q^{7/2}-14 q^{5/2}+15 q^{3/2}-15 \sqrt{q}+\frac{11}{\sqrt{q}}-\frac{8}{q^{3/2}}+\frac{4}{q^{5/2}}-\frac{1}{q^{7/2}} (db)
Signature 1 (db)
HOMFLY-PT polynomial -z^3 a^{-5} -2 z a^{-5} - a^{-5} z^{-1} +2 z^5 a^{-3} +6 z^3 a^{-3} +6 z a^{-3} +2 a^{-3} z^{-1} -z^7 a^{-1} +a z^5-4 z^5 a^{-1} +2 a z^3-6 z^3 a^{-1} +a z-4 z a^{-1} +a z^{-1} -2 a^{-1} z^{-1} (db)
Kauffman polynomial z^5 a^{-7} -2 z^3 a^{-7} +z a^{-7} +3 z^6 a^{-6} -5 z^4 a^{-6} +2 z^2 a^{-6} +5 z^7 a^{-5} -8 z^5 a^{-5} +5 z^3 a^{-5} -3 z a^{-5} + a^{-5} z^{-1} +5 z^8 a^{-4} -6 z^6 a^{-4} +3 z^4 a^{-4} -2 z^2 a^{-4} +2 z^9 a^{-3} +8 z^7 a^{-3} +a^3 z^5-23 z^5 a^{-3} -a^3 z^3+23 z^3 a^{-3} -11 z a^{-3} +2 a^{-3} z^{-1} +11 z^8 a^{-2} +4 a^2 z^6-20 z^6 a^{-2} -6 a^2 z^4+14 z^4 a^{-2} +a^2 z^2-5 z^2 a^{-2} + a^{-2} +2 z^9 a^{-1} +7 a z^7+10 z^7 a^{-1} -13 a z^5-28 z^5 a^{-1} +7 a z^3+24 z^3 a^{-1} -2 a z-9 z a^{-1} +a z^{-1} +2 a^{-1} z^{-1} +6 z^8-7 z^6 (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        51 4
8       62  -4
6      85   3
4     87    -1
2    77     0
0   59      4
-2  36       -3
-4 15        4
-6 3         -3
-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}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=0 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{7}
r=1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{8}
r=3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
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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L10a68

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L10a70