L10a127

L10a126

L10a128

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Link Presentations

[edit Notes on L10a127's Link Presentations]

Planar diagram presentation	X6172 X12,6,13,5 X8493 X2,14,3,13 X14,7,15,8 X18,10,19,9 X16,12,17,11 X20,18,11,17 X4,15,1,16 X10,20,5,19
Gauss code	{1, -4, 3, -9}, {2, -1, 5, -3, 6, -10}, {7, -2, 4, -5, 9, -7, 8, -6, 10, -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{(u-1) (v-1) (w-1) (2 v w-v-w+2)}{\sqrt{u} v w} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^8+4 q^7-8 q^6+12 q^5-15 q^4+17 q^3-14 q^2+13 q-7+4 q^{-1} - q^{-2} }[/math] (db)
Signature	2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^6 a^{-2} +z^6 a^{-4} +2 z^4 a^{-2} +2 z^4 a^{-4} -z^4 a^{-6} -z^4+z^2 a^{-2} +z^2 a^{-4} -z^2 a^{-6} -z^2- a^{-2} +1-2 a^{-2} z^{-2} + a^{-4} z^{-2} + z^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ 2 z^9 a^{-3} +2 z^9 a^{-5} +5 z^8 a^{-2} +11 z^8 a^{-4} +6 z^8 a^{-6} +6 z^7 a^{-1} +10 z^7 a^{-3} +11 z^7 a^{-5} +7 z^7 a^{-7} -13 z^6 a^{-4} -5 z^6 a^{-6} +4 z^6 a^{-8} +4 z^6+a z^5-8 z^5 a^{-1} -20 z^5 a^{-3} -24 z^5 a^{-5} -12 z^5 a^{-7} +z^5 a^{-9} -11 z^4 a^{-2} -z^4 a^{-4} -3 z^4 a^{-6} -6 z^4 a^{-8} -7 z^4-a z^3+2 z^3 a^{-1} +7 z^3 a^{-3} +11 z^3 a^{-5} +6 z^3 a^{-7} -z^3 a^{-9} +6 z^2 a^{-2} +2 z^2 a^{-4} +2 z^2 a^{-6} +2 z^2 a^{-8} +4 z^2-z a^{-1} -z a^{-3} + a^{-2} + a^{-4} +1+2 a^{-1} z^{-1} +2 a^{-3} z^{-1} -2 a^{-2} z^{-2} - a^{-4} 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   \ -3 -2 -1 0 1 2 3 4 5 6 7 χ 17                     1 -1 15                   3   3 13                 5 1   -4 11               7 3     4 9             8 5       -3 7           9 7         2 5         7 10           3 3       6 7             -1 1     3 9               6 -1   1 4                 -3 -3   3                   3 -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}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\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^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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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