L10a47

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L10a46.gif

L10a46

L10a48.gif

L10a48

Contents

L10a47.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L10a47's Link Presentations]

Planar diagram presentation X6172 X14,3,15,4 X18,8,19,7 X20,10,5,9 X8,20,9,19 X16,12,17,11 X12,16,13,15 X10,18,11,17 X2536 X4,13,1,14
Gauss code {1, -9, 2, -10}, {9, -1, 3, -5, 4, -8, 6, -7, 10, -2, 7, -6, 8, -3, 5, -4}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gif
BraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L10a47 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{2 u v^3-4 u v^2+4 u v-3 u-3 v^3+4 v^2-4 v+2}{\sqrt{u} v^{3/2}} (db)
Jones polynomial -5 q^{9/2}+7 q^{7/2}-\frac{1}{q^{7/2}}-8 q^{5/2}+\frac{1}{q^{5/2}}+9 q^{3/2}-\frac{4}{q^{3/2}}-q^{13/2}+3 q^{11/2}-8 \sqrt{q}+\frac{5}{\sqrt{q}} (db)
Signature 1 (db)
HOMFLY-PT polynomial z^5 a^{-1} +z^5 a^{-3} -2 a z^3+2 z^3 a^{-1} +2 z^3 a^{-3} -z^3 a^{-5} +a^3 z-5 a z+2 z a^{-1} +z a^{-3} -z a^{-5} +2 a^3 z^{-1} -3 a z^{-1} + a^{-1} z^{-1} (db)
Kauffman polynomial -z^9 a^{-1} -z^9 a^{-3} -4 z^8 a^{-2} -3 z^8 a^{-4} -z^8-a z^7+2 z^7 a^{-1} -z^7 a^{-3} -4 z^7 a^{-5} -a^2 z^6+11 z^6 a^{-2} +7 z^6 a^{-4} -3 z^6 a^{-6} -a^3 z^5-a z^5-7 z^5 a^{-1} +4 z^5 a^{-3} +10 z^5 a^{-5} -z^5 a^{-7} +a^2 z^4-16 z^4 a^{-2} -7 z^4 a^{-4} +7 z^4 a^{-6} -z^4+4 a^3 z^3+7 a z^3+7 z^3 a^{-1} -4 z^3 a^{-3} -6 z^3 a^{-5} +2 z^3 a^{-7} +3 a^2 z^2+7 z^2 a^{-2} +3 z^2 a^{-4} -2 z^2 a^{-6} +5 z^2-5 a^3 z-7 a z-3 z a^{-1} +z a^{-5} -3 a^2- a^{-2} -3+2 a^3 z^{-1} +3 a z^{-1} + a^{-1} 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       42  -2
6      43   1
4     54    -1
2    34     -1
0   36      3
-2  12       -1
-4  3        3
-611         0
-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} {\mathbb Z}
r=-3 {\mathbb Z}
r=-2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=0 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
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

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See/edit the Link Page master template (intermediate).

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L10a48