L10a104

L10a103

L10a105

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Link Presentations

[edit Notes on L10a104's Link Presentations]

Planar diagram presentation	X10,1,11,2 X16,11,17,12 X8,9,1,10 X20,17,9,18 X12,4,13,3 X14,8,15,7 X6,14,7,13 X18,6,19,5 X4,20,5,19 X2,16,3,15
Gauss code	{1, -10, 5, -9, 8, -7, 6, -3}, {3, -1, 2, -5, 7, -6, 10, -2, 4, -8, 9, -4}

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^3 \left(-v^3\right)+2 u^3 v^2-u^3 v+3 u^2 v^3-7 u^2 v^2+7 u^2 v-2 u^2-2 u v^3+7 u v^2-7 u v+3 u-v^2+2 v-1}{u^{3/2} v^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -7 q^{9/2}+11 q^{7/2}-\frac{1}{q^{7/2}}-15 q^{5/2}+\frac{4}{q^{5/2}}+15 q^{3/2}-\frac{8}{q^{3/2}}-q^{13/2}+3 q^{11/2}-15 \sqrt{q}+\frac{12}{\sqrt{q}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -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} + 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-3 z a^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ 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} +6 z^3 a^{-5} -4 z a^{-5} + a^{-5} z^{-1} +5 z^8 a^{-4} -5 z^6 a^{-4} +2 z^2 a^{-4} - a^{-4} +2 z^9 a^{-3} +9 z^7 a^{-3} +a^3 z^5-25 z^5 a^{-3} -a^3 z^3+23 z^3 a^{-3} -9 z a^{-3} + a^{-3} z^{-1} +11 z^8 a^{-2} +4 a^2 z^6-17 z^6 a^{-2} -6 a^2 z^4+6 z^4 a^{-2} +2 a^2 z^2+z^2 a^{-2} +2 z^9 a^{-1} +7 a z^7+11 z^7 a^{-1} -12 a z^5-29 z^5 a^{-1} +6 a z^3+22 z^3 a^{-1} -2 a z-6 z a^{-1} +6 z^8-5 z^6-5 z^4+3 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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 χ 14                     1 1 12                   2   -2 10                 5 1   4 8               7 3     -4 6             8 4       4 4           7 7         0 2         8 8           0 0       5 8             3 -2     3 7               -4 -4   1 5                 4 -6   3                   -3 -8 1                     1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=0 }[/math] [math]\displaystyle{ i=2 }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=6 }[/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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