L11a535

L11a534

L11a536

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Link Presentations

[edit Notes on L11a535's Link Presentations]

Planar diagram presentation	X6172 X2536 X18,12,19,11 X10,3,11,4 X4,9,1,10 X16,7,17,8 X8,15,5,16 X22,18,15,17 X20,14,21,13 X12,20,13,19 X14,22,9,21
Gauss code	{1, -2, 4, -5}, {2, -1, 6, -7}, {5, -4, 3, -10, 9, -11}, {7, -6, 8, -3, 10, -9, 11, -8}

A Braid Representative	{{{braid_table}}}
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{-2 t(1) t(4)^2 t(3)^2+t(1) t(2) t(4)^2 t(3)^2-2 t(2) t(4)^2 t(3)^2+2 t(4)^2 t(3)^2+t(1) t(4) t(3)^2-t(1) t(2) t(4) t(3)^2+2 t(2) t(4) t(3)^2-t(4) t(3)^2+2 t(1) t(4)^2 t(3)-t(1) t(2) t(4)^2 t(3)+t(2) t(4)^2 t(3)-t(4)^2 t(3)+t(1) t(3)-t(1) t(2) t(3)+2 t(2) t(3)-3 t(1) t(4) t(3)+2 t(1) t(2) t(4) t(3)-3 t(2) t(4) t(3)+2 t(4) t(3)-t(3)-2 t(1)+2 t(1) t(2)-2 t(2)+2 t(1) t(4)-t(1) t(2) t(4)+t(2) t(4)-t(4)+1}{\sqrt{t(1)} \sqrt{t(2)} t(3) t(4)} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -4 q^{9/2}+\frac{1}{q^{9/2}}+7 q^{7/2}-\frac{5}{q^{7/2}}-11 q^{5/2}+\frac{6}{q^{5/2}}+13 q^{3/2}-\frac{12}{q^{3/2}}+q^{11/2}-\frac{1}{q^{11/2}}-15 \sqrt{q}+\frac{12}{\sqrt{q}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^7 a^{-1} +3 a z^5-4 z^5 a^{-1} +z^5 a^{-3} -3 a^3 z^3+12 a z^3-6 z^3 a^{-1} +2 z^3 a^{-3} +a^5 z-11 a^3 z+16 a z-6 z a^{-1} +3 a^5 z^{-1} -10 a^3 z^{-1} +11 a z^{-1} -4 a^{-1} z^{-1} +a^5 z^{-3} -3 a^3 z^{-3} +3 a z^{-3} - a^{-1} z^{-3} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -a^2 z^{10}-z^{10}-a^3 z^9-5 a z^9-4 z^9 a^{-1} -a^4 z^8+a^2 z^8-7 z^8 a^{-2} -5 z^8-a^5 z^7-a^3 z^7+12 a z^7+4 z^7 a^{-1} -8 z^7 a^{-3} +2 a^4 z^6-2 a^2 z^6+10 z^6 a^{-2} -7 z^6 a^{-4} +13 z^6+6 a^5 z^5+17 a^3 z^5-2 a z^5+9 z^5 a^{-3} -4 z^5 a^{-5} +6 a^4 z^4+19 a^2 z^4-4 z^4 a^{-2} +7 z^4 a^{-4} -z^4 a^{-6} +z^4-13 a^5 z^3-32 a^3 z^3-16 a z^3+3 z^3 a^{-5} -17 a^4 z^2-33 a^2 z^2-16 z^2+13 a^5 z+28 a^3 z+21 a z+3 z a^{-1} -3 z a^{-3} +13 a^4+24 a^2- a^{-2} +11-6 a^5 z^{-1} -14 a^3 z^{-1} -12 a z^{-1} -3 a^{-1} z^{-1} + a^{-3} z^{-1} -3 a^4 z^{-2} -6 a^2 z^{-2} -3 z^{-2} +a^5 z^{-3} +3 a^3 z^{-3} +3 a z^{-3} + a^{-1} z^{-3} }[/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                     3   3 8                   4 1   -3 6                 7 3     4 4               9 7       -2 2             6 4         2 0           7 10           3 -2         5 5             0 -4       2 8               6 -6     3 4                 -1 -8   1 5                   4 -10                         0 -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}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\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^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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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