L10a152

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L10a151.gif

L10a151

L10a153.gif

L10a153

Contents

L10a152.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L10a152's Link Presentations]

Planar diagram presentation X6172 X10,3,11,4 X20,14,15,13 X16,8,17,7 X8,16,9,15 X18,12,19,11 X12,20,13,19 X14,18,5,17 X2536 X4,9,1,10
Gauss code {1, -9, 2, -10}, {5, -4, 8, -6, 7, -3}, {9, -1, 4, -5, 10, -2, 6, -7, 3, -8}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gif
BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L10a152 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(t(3)-1) \left(-t(2) t(3)^3+t(1) t(2)^2 t(3)^2-2 t(2)^2 t(3)^2-2 t(1) t(2) t(3)^2+2 t(2) t(3)^2-t(3)^2-t(1) t(2)^2 t(3)-2 t(1) t(3)+2 t(1) t(2) t(3)-2 t(2) t(3)+t(3)-t(1) t(2)\right)}{\sqrt{t(1)} t(2) t(3)^2} (db)
Jones polynomial q^7-3 q^6+7 q^5-9 q^4+12 q^3+ q^{-3} -12 q^2-2 q^{-2} +11 q+6 q^{-1} -8 (db)
Signature 2 (db)
HOMFLY-PT polynomial z^2 a^{-6} + a^{-6} -2 z^4 a^{-4} -3 z^2 a^{-4} + a^{-4} z^{-2} + a^{-4} +z^6 a^{-2} +2 z^4 a^{-2} +a^2 z^2-z^2 a^{-2} -2 a^{-2} z^{-2} +2 a^2-4 a^{-2} -2 z^4-4 z^2+ z^{-2} (db)
Kauffman polynomial z^4 a^{-8} -z^2 a^{-8} +3 z^5 a^{-7} -2 z^3 a^{-7} +6 z^6 a^{-6} -9 z^4 a^{-6} +8 z^2 a^{-6} -3 a^{-6} +6 z^7 a^{-5} -6 z^5 a^{-5} +z^3 a^{-5} +z a^{-5} +4 z^8 a^{-4} -7 z^4 a^{-4} +3 z^2 a^{-4} - a^{-4} z^{-2} +2 a^{-4} +z^9 a^{-3} +10 z^7 a^{-3} -27 z^5 a^{-3} +24 z^3 a^{-3} -10 z a^{-3} +2 a^{-3} z^{-1} +7 z^8 a^{-2} +a^2 z^6-15 z^6 a^{-2} -4 a^2 z^4+15 z^4 a^{-2} +5 a^2 z^2-18 z^2 a^{-2} -2 a^{-2} z^{-2} -2 a^2+10 a^{-2} +z^9 a^{-1} +2 a z^7+6 z^7 a^{-1} -5 a z^5-23 z^5 a^{-1} +2 a z^3+23 z^3 a^{-1} +a z-10 z a^{-1} +2 a^{-1} z^{-1} +3 z^8-8 z^6+8 z^4-7 z^2- z^{-2} +4 (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
15          11
13         31-2
11        4  4
9       53  -2
7      74   3
5     66    0
3    56     -1
1   47      3
-1  24       -2
-3  4        4
-512         -1
-71          1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=1 i=3
r=-4 {\mathbb Z} {\mathbb Z}
r=-3 {\mathbb Z}^{2}
r=-2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=0 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{5}
r=1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{7}
r=3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=5 {\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=6 {\mathbb Z} {\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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L10a151

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L10a153