L10a149

L10a148

L10a150

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Link Presentations

[edit Notes on L10a149's Link Presentations]

Planar diagram presentation	X6172 X12,7,13,8 X4,13,1,14 X16,6,17,5 X8493 X20,18,15,17 X10,20,11,19 X18,10,19,9 X14,16,5,15 X2,12,3,11
Gauss code	{1, -10, 5, -3}, {9, -4, 6, -8, 7, -6}, {4, -1, 2, -5, 8, -7, 10, -2, 3, -9}

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(1)-1) (t(3)-1)^2 (2 t(3) t(2)-t(2)-t(3)+2)}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^8+5 q^7-9 q^6+13 q^5-15 q^4+17 q^3-14 q^2+12 q-6+3 q^{-1} - q^{-2} }[/math] (db)
Signature	2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^4 a^{-6} + a^{-6} z^{-2} +2 a^{-6} +z^6 a^{-4} +z^4 a^{-4} -4 z^2 a^{-4} -2 a^{-4} z^{-2} -7 a^{-4} +z^6 a^{-2} +3 z^4 a^{-2} +6 z^2 a^{-2} + a^{-2} z^{-2} +6 a^{-2} -z^4-2 z^2-1 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^5 a^{-9} +5 z^6 a^{-8} -6 z^4 a^{-8} + a^{-8} +9 z^7 a^{-7} -15 z^5 a^{-7} +4 z^3 a^{-7} +7 z^8 a^{-6} -5 z^6 a^{-6} -8 z^4 a^{-6} +5 z^2 a^{-6} + a^{-6} z^{-2} -3 a^{-6} +2 z^9 a^{-5} +13 z^7 a^{-5} -28 z^5 a^{-5} +9 z^3 a^{-5} +5 z a^{-5} -2 a^{-5} z^{-1} +11 z^8 a^{-4} -12 z^6 a^{-4} -9 z^4 a^{-4} +18 z^2 a^{-4} +2 a^{-4} z^{-2} -10 a^{-4} +2 z^9 a^{-3} +8 z^7 a^{-3} -16 z^5 a^{-3} +4 z^3 a^{-3} +7 z a^{-3} -2 a^{-3} z^{-1} +4 z^8 a^{-2} +z^6 a^{-2} -13 z^4 a^{-2} +18 z^2 a^{-2} + a^{-2} z^{-2} -9 a^{-2} +4 z^7 a^{-1} +a z^5-3 z^5 a^{-1} -2 a z^3-3 z^3 a^{-1} +a z+3 z a^{-1} +3 z^6-6 z^4+5 z^2-2 }[/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   \ -3 -2 -1 0 1 2 3 4 5 6 7 χ 17                     1 -1 15                   4   4 13                 5 1   -4 11               8 4     4 9             9 7       -2 7           8 6         2 5         6 9           3 3       6 8             -2 1     2 8               6 -1   1 4                 -3 -3   2                   2 -5 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=-3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=7 }[/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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