L11a544

L11a543

L11a545

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Link Presentations

[edit Notes on L11a544's Link Presentations]

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

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(3) t(2)^2-2 t(3) t(2)^2+2 t(1) t(3)^2 t(4) t(2)^2-2 t(3)^2 t(4) t(2)^2-t(1) t(3) t(4) t(2)^2+3 t(3) t(4) t(2)^2-t(4) t(2)^2+t(2)^2+2 t(1) t(3)^2 t(2)-t(3)^2 t(2)+t(1) t(2)-3 t(1) t(3) t(2)+3 t(3) t(2)-3 t(1) t(3)^2 t(4) t(2)+t(3)^2 t(4) t(2)-t(1) t(4) t(2)+3 t(1) t(3) t(4) t(2)-3 t(3) t(4) t(2)+2 t(4) t(2)-3 t(2)-t(1) t(3)^2-2 t(1)+3 t(1) t(3)-t(3)+t(1) t(3)^2 t(4)+t(1) t(4)-2 t(1) t(3) t(4)+t(3) t(4)-t(4)+2}{\sqrt{t(1)} t(2) t(3) \sqrt{t(4)}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{23/2}-4 q^{21/2}+8 q^{19/2}-13 q^{17/2}+16 q^{15/2}-19 q^{13/2}+16 q^{11/2}-16 q^{9/2}+9 q^{7/2}-7 q^{5/2}+2 q^{3/2}-\sqrt{q} }[/math] (db)
Signature	5 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^5 a^{-9} +2 z^3 a^{-9} - a^{-9} z^{-3} - a^{-9} z^{-1} -z^7 a^{-7} -3 z^5 a^{-7} -z^3 a^{-7} +3 a^{-7} z^{-3} +4 z a^{-7} +6 a^{-7} z^{-1} -z^7 a^{-5} -4 z^5 a^{-5} -7 z^3 a^{-5} -3 a^{-5} z^{-3} -10 z a^{-5} -9 a^{-5} z^{-1} +z^5 a^{-3} +4 z^3 a^{-3} + a^{-3} z^{-3} +6 z a^{-3} +4 a^{-3} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^4 a^{-14} +4 z^5 a^{-13} -2 z^3 a^{-13} +8 z^6 a^{-12} -7 z^4 a^{-12} +z^2 a^{-12} +11 z^7 a^{-11} -15 z^5 a^{-11} +8 z^3 a^{-11} -2 z a^{-11} +10 z^8 a^{-10} -13 z^6 a^{-10} +2 z^4 a^{-10} +2 z^2 a^{-10} +5 z^9 a^{-9} +5 z^7 a^{-9} -32 z^5 a^{-9} +32 z^3 a^{-9} - a^{-9} z^{-3} -16 z a^{-9} +5 a^{-9} z^{-1} +z^{10} a^{-8} +14 z^8 a^{-8} -37 z^6 a^{-8} +17 z^4 a^{-8} +10 z^2 a^{-8} +3 a^{-8} z^{-2} -10 a^{-8} +7 z^9 a^{-7} -7 z^7 a^{-7} -28 z^5 a^{-7} +48 z^3 a^{-7} -3 a^{-7} z^{-3} -31 z a^{-7} +12 a^{-7} z^{-1} +z^{10} a^{-6} +6 z^8 a^{-6} -22 z^6 a^{-6} +9 z^4 a^{-6} +18 z^2 a^{-6} +6 a^{-6} z^{-2} -19 a^{-6} +2 z^9 a^{-5} -20 z^5 a^{-5} +36 z^3 a^{-5} -3 a^{-5} z^{-3} -27 z a^{-5} +12 a^{-5} z^{-1} +2 z^8 a^{-4} -6 z^6 a^{-4} +2 z^4 a^{-4} +9 z^2 a^{-4} +3 a^{-4} z^{-2} -10 a^{-4} +z^7 a^{-3} -5 z^5 a^{-3} +10 z^3 a^{-3} - a^{-3} z^{-3} -10 z a^{-3} +5 a^{-3} z^{-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   \ -2 -1 0 1 2 3 4 5 6 7 8 9 χ 24                       1 -1 22                     3   3 20                   5 1   -4 18                 8 3     5 16               10 7       -3 14             9 6         3 12           9 12           3 10         7 7             0 8       4 11               7 6     3 5                 -2 4   1 6                   5 2   1                     -1 0 1                       1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=4 }[/math] [math]\displaystyle{ i=6 }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=7 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=8 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=9 }[/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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