L11n455

L11n454

L11n456

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Link Presentations

[edit Notes on L11n455's Link Presentations]

Planar diagram presentation	X6172 X14,5,15,6 X12,4,13,3 X2,9,3,10 X7,19,8,18 X17,9,18,8 X10,13,5,14 X19,22,20,17 X21,11,22,16 X11,21,12,20 X4,16,1,15
Gauss code	{1, -4, 3, -11}, {2, -1, -5, 6, 4, -7}, {-10, -3, 7, -2, 11, 9}, {-6, 5, -8, 10, -9, 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(1) t(3)^2 t(2)^2+t(3)^2 t(2)^2-t(1) t(2)^2+2 t(1) t(3) t(2)^2+t(1) t(4) t(2)^2-t(1) t(3) t(4) t(2)^2-2 t(3)^2 t(2)+t(1) t(2)-t(1) t(3) t(2)+t(3) t(2)+t(3)^2 t(4) t(2)-2 t(1) t(4) t(2)+t(1) t(3) t(4) t(2)-t(3) t(4) t(2)+t(3)^2-t(3)-t(3)^2 t(4)+t(1) t(4)+2 t(3) t(4)-t(4)}{\sqrt{t(1)} t(2) t(3) \sqrt{t(4)}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 2 q^{9/2}-6 q^{7/2}+\frac{1}{q^{7/2}}+5 q^{5/2}-\frac{4}{q^{5/2}}-9 q^{3/2}+\frac{5}{q^{3/2}}-q^{11/2}+7 \sqrt{q}-\frac{8}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -2 z^5 a^{-1} +3 a z^3-8 z^3 a^{-1} +3 z^3 a^{-3} -a^3 z+6 a z-12 z a^{-1} +8 z a^{-3} -z a^{-5} +3 a z^{-1} -8 a^{-1} z^{-1} +7 a^{-3} z^{-1} -2 a^{-5} z^{-1} +a z^{-3} -3 a^{-1} z^{-3} +3 a^{-3} z^{-3} - a^{-5} z^{-3} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^9 a^{-1} -z^9 a^{-3} -6 z^8 a^{-2} -2 z^8 a^{-4} -4 z^8-5 a z^7-7 z^7 a^{-1} -3 z^7 a^{-3} -z^7 a^{-5} -2 a^2 z^6+19 z^6 a^{-2} +7 z^6 a^{-4} +10 z^6+17 a z^5+39 z^5 a^{-1} +27 z^5 a^{-3} +5 z^5 a^{-5} +2 a^2 z^4-6 z^4 a^{-2} -3 z^4 a^{-4} -z^4-4 a^3 z^3-24 a z^3-52 z^3 a^{-1} -42 z^3 a^{-3} -10 z^3 a^{-5} -a^4 z^2-2 a^2 z^2-18 z^2 a^{-2} -9 z^2 a^{-4} -10 z^2+2 a^3 z+16 a z+31 z a^{-1} +27 z a^{-3} +10 z a^{-5} +19 a^{-2} +10 a^{-4} +10-5 a z^{-1} -12 a^{-1} z^{-1} -12 a^{-3} z^{-1} -5 a^{-5} z^{-1} -6 a^{-2} z^{-2} -3 a^{-4} z^{-2} -3 z^{-2} +a z^{-3} +3 a^{-1} z^{-3} +3 a^{-3} z^{-3} + a^{-5} 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   \ -3 -2 -1 0 1 2 3 4 5 6 χ 12                   1 1 10                 1   -1 8               5 1   4 6             2 3     1 4           7 3       4 2       1 3 4         2 0       6 5           1 -2   1 2 4             3 -4   3 4               -1 -6   3                 3 -8 1                   -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-2 }[/math] [math]\displaystyle{ i=0 }[/math] [math]\displaystyle{ i=2 }[/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}^{3} }[/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}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=6 }[/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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