L11a328

L11a327

L11a329

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Link Presentations

[edit Notes on L11a328's Link Presentations]

Planar diagram presentation	X10,1,11,2 X12,4,13,3 X20,5,21,6 X18,9,19,10 X22,19,9,20 X16,12,17,11 X6,21,7,22 X14,8,15,7 X4,14,5,13 X8,16,1,15 X2,17,3,18
Gauss code	{1, -11, 2, -9, 3, -7, 8, -10}, {4, -1, 6, -2, 9, -8, 10, -6, 11, -4, 5, -3, 7, -5}

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(2)-1) \left(2 t(2)^2 t(1)^2-t(2) t(1)^2-2 t(2)^2 t(1)+4 t(2) t(1)-2 t(1)-t(2)+2\right)}{t(1)^{3/2} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -3 q^{9/2}+\frac{4}{q^{9/2}}+6 q^{7/2}-\frac{8}{q^{7/2}}-11 q^{5/2}+\frac{12}{q^{5/2}}+15 q^{3/2}-\frac{16}{q^{3/2}}+q^{11/2}-\frac{1}{q^{11/2}}-18 \sqrt{q}+\frac{17}{\sqrt{q}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^3 z^5+z^5 a^{-3} +2 a^3 z^3+3 z^3 a^{-3} +2 z a^{-3} -a z^7-z^7 a^{-1} -3 a z^5-4 z^5 a^{-1} -a z^3-6 z^3 a^{-1} +2 a z-4 z a^{-1} +a z^{-1} - a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^4 a^{-6} -z^2 a^{-6} +a^5 z^7-3 a^5 z^5+3 z^5 a^{-5} +3 a^5 z^3-3 z^3 a^{-5} -a^5 z+z a^{-5} +4 a^4 z^8-14 a^4 z^6+5 z^6 a^{-4} +14 a^4 z^4-4 z^4 a^{-4} -4 a^4 z^2+z^2 a^{-4} +5 a^3 z^9-14 a^3 z^7+7 z^7 a^{-3} +5 a^3 z^5-8 z^5 a^{-3} +6 a^3 z^3+4 z^3 a^{-3} -3 a^3 z-z a^{-3} +2 a^2 z^{10}+6 a^2 z^8+8 z^8 a^{-2} -37 a^2 z^6-13 z^6 a^{-2} +38 a^2 z^4+8 z^4 a^{-2} -10 a^2 z^2-z^2 a^{-2} +11 a z^9+6 z^9 a^{-1} -28 a z^7-6 z^7 a^{-1} +9 a z^5-10 z^5 a^{-1} +11 a z^3+15 z^3 a^{-1} -6 a z-6 z a^{-1} +a z^{-1} + a^{-1} z^{-1} +2 z^{10}+10 z^8-41 z^6+37 z^4-9 z^2-1 }[/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 12                       1 -1 10                     2   2 8                   4 1   -3 6                 7 2     5 4               8 4       -4 2             10 7         3 0           9 10           1 -2         7 8             -1 -4       5 9               4 -6     3 7                 -4 -8   1 5                   4 -10   3                     -3 -12 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=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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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