L10a138

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L10a137.gif

L10a137

L10a139.gif

L10a139

Contents

L10a138.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L10a138's Link Presentations]

Planar diagram presentation X6172 X14,6,15,5 X8493 X2,16,3,15 X16,7,17,8 X18,10,19,9 X20,12,13,11 X12,14,5,13 X4,17,1,18 X10,20,11,19
Gauss code {1, -4, 3, -9}, {2, -1, 5, -3, 6, -10, 7, -8}, {8, -2, 4, -5, 9, -6, 10, -7}
A Braid Representative
BraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gif
BraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gif
A Morse Link Presentation L10a138 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(t(1)-1) (t(2)-1) (t(3)-1) \left(t(2)^2 t(3)^2+t(2) t(3)+1\right)}{\sqrt{t(1)} t(2)^{3/2} t(3)^{3/2}} (db)
Jones polynomial -q^9+3 q^8-4 q^7+6 q^6-7 q^5+8 q^4-6 q^3+6 q^2-3 q+3- q^{-1} (db)
Signature 4 (db)
HOMFLY-PT polynomial -z^6 a^{-6} -4 z^4 a^{-6} -3 z^2 a^{-6} + a^{-6} z^{-2} + a^{-6} +z^8 a^{-4} +6 z^6 a^{-4} +11 z^4 a^{-4} +5 z^2 a^{-4} -2 a^{-4} z^{-2} -4 a^{-4} -z^6 a^{-2} -4 z^4 a^{-2} -2 z^2 a^{-2} + a^{-2} z^{-2} +3 a^{-2} (db)
Kauffman polynomial z^3 a^{-11} +3 z^4 a^{-10} -2 z^2 a^{-10} +4 z^5 a^{-9} -3 z^3 a^{-9} +4 z^6 a^{-8} -3 z^4 a^{-8} -2 z^2 a^{-8} + a^{-8} +4 z^7 a^{-7} -6 z^5 a^{-7} +4 z^8 a^{-6} -12 z^6 a^{-6} +12 z^4 a^{-6} -6 z^2 a^{-6} + a^{-6} z^{-2} - a^{-6} +2 z^9 a^{-5} -4 z^7 a^{-5} -3 z^5 a^{-5} +3 z^3 a^{-5} +4 z a^{-5} -2 a^{-5} z^{-1} +7 z^8 a^{-4} -31 z^6 a^{-4} +39 z^4 a^{-4} -13 z^2 a^{-4} +2 a^{-4} z^{-2} -4 a^{-4} +2 z^9 a^{-3} -7 z^7 a^{-3} +3 z^5 a^{-3} +2 z^3 a^{-3} +4 z a^{-3} -2 a^{-3} z^{-1} +3 z^8 a^{-2} -15 z^6 a^{-2} +21 z^4 a^{-2} -7 z^2 a^{-2} + a^{-2} z^{-2} -3 a^{-2} +z^7 a^{-1} -4 z^5 a^{-1} +3 z^3 a^{-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 \
-3-2-101234567χ
19          1-1
17         2 2
15        21 -1
13       42  2
11      54   -1
9     32    1
7    35     2
5   33      0
3  25       3
1 11        0
-1 2         2
-31          -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=3 i=5
r=-3 {\mathbb Z}
r=-2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=0 {\mathbb Z}^{5}\oplus{\mathbb Z}_2 {\mathbb Z}^{3}
r=1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{4}
r=5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=6 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=7 {\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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L10a137.gif

L10a137

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L10a139