L10a152

L10a151

L10a153

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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
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{(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} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 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 }[/math] (db)
Signature	2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ 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} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ 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 }[/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 15                     1 1 13                   3 1 -2 11                 4     4 9               5 3     -2 7             7 4       3 5           6 6         0 3         5 6           -1 1       4 7             3 -1     2 4               -2 -3     4                 4 -5 1 2                   -1 -7 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=3 }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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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