L11a520

L11a519

L11a521

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Link Presentations

[edit Notes on L11a520's Link Presentations]

Planar diagram presentation	X8192 X18,8,19,7 X16,6,17,5 X10,16,11,15 X14,3,15,4 X4,11,5,12 X2,20,3,19 X20,9,21,10 X12,13,7,14 X22,18,13,17 X6,21,1,22
Gauss code	{1, -7, 5, -6, 3, -11}, {2, -1, 8, -4, 6, -9}, {9, -5, 4, -3, 10, -2, 7, -8, 11, -10}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in [math]\displaystyle{ u }[/math], [math]\displaystyle{ v }[/math], [math]\displaystyle{ w }[/math], ...)	[math]\displaystyle{ \frac{(u-1) (v-1) (w-1) \left(u v w^2-u v w+u v-u w^2+2 u w-u-v w^2+2 v w-v+w^2-w+1\right)}{u v w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^5+5 q^4-13 q^3+23 q^2-31 q+37-36 q^{-1} +33 q^{-2} -23 q^{-3} +15 q^{-4} -6 q^{-5} + q^{-6} }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^4 z^4+a^4 z^{-2} -2 a^2 z^6-z^6 a^{-2} -4 a^2 z^4-2 z^4 a^{-2} -2 a^2 z^2-2 z^2 a^{-2} -2 a^2 z^{-2} -a^2+z^8+4 z^6+7 z^4+4 z^2+ z^{-2} +1 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ 7 a^2 z^{10}+7 z^{10}+17 a^3 z^9+36 a z^9+19 z^9 a^{-1} +15 a^4 z^8+22 a^2 z^8+21 z^8 a^{-2} +28 z^8+6 a^5 z^7-27 a^3 z^7-67 a z^7-21 z^7 a^{-1} +13 z^7 a^{-3} +a^6 z^6-27 a^4 z^6-72 a^2 z^6-32 z^6 a^{-2} +5 z^6 a^{-4} -81 z^6-6 a^5 z^5+6 a^3 z^5+26 a z^5-13 z^5 a^{-3} +z^5 a^{-5} +11 a^4 z^4+41 a^2 z^4+21 z^4 a^{-2} -2 z^4 a^{-4} +53 z^4+a^3 z^3-a z^3+3 z^3 a^{-1} +5 z^3 a^{-3} +2 a^4 z^2-2 a^2 z^2-6 z^2 a^{-2} -10 z^2-a^3 z-a z+a^4+a^2+1+2 a^3 z^{-1} +2 a z^{-1} -a^4 z^{-2} -2 a^2 z^{-2} - z^{-2} }[/math] (db)

Khovanov Homology

The coefficients of the monomials [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]).

\   r    \    j   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 11                       1 -1 9                     4   4 7                   9 1   -8 5                 14 4     10 3               17 9       -8 1             20 14         6 -1           18 19           1 -3         15 18             -3 -5       10 20               10 -7     5 13                 -8 -9   1 10                   9 -11   5                     -5 -13 1                       1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-1 }[/math] [math]\displaystyle{ i=1 }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{20}\oplus{\mathbb Z}_2^{13} }[/math] [math]\displaystyle{ {\mathbb Z}^{15} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{18}\oplus{\mathbb Z}_2^{18} }[/math] [math]\displaystyle{ {\mathbb Z}^{18} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{19}\oplus{\mathbb Z}_2^{18} }[/math] [math]\displaystyle{ {\mathbb Z}^{20} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{17} }[/math] [math]\displaystyle{ {\mathbb Z}^{17} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{14} }[/math] [math]\displaystyle{ {\mathbb Z}^{14} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]

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). See/edit the Link_Splice_Base (expert). Back to the top.

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