L11a530

L11a529

L11a531

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Link Presentations

[edit Notes on L11a530's Link Presentations]

Planar diagram presentation	X8192 X12,3,13,4 X20,14,21,13 X18,10,19,9 X10,22,11,21 X14,20,7,19 X16,5,17,6 X22,18,15,17 X2738 X4,11,5,12 X6,15,1,16
Gauss code	{1, -9, 2, -10, 7, -11}, {9, -1, 4, -5, 10, -2, 3, -6}, {11, -7, 8, -4, 6, -3, 5, -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(2)^3-t(1) t(3) t(2)^3+2 t(3) t(2)^3-t(2)^3+2 t(1)^2 t(2)^2+t(1)^2 t(3)^2 t(2)^2-3 t(1) t(3)^2 t(2)^2+3 t(3)^2 t(2)^2-4 t(1) t(2)^2-3 t(1)^2 t(3) t(2)^2+8 t(1) t(3) t(2)^2-5 t(3) t(2)^2+2 t(2)^2-3 t(1)^2 t(2)-2 t(1)^2 t(3)^2 t(2)+4 t(1) t(3)^2 t(2)-2 t(3)^2 t(2)+3 t(1) t(2)+5 t(1)^2 t(3) t(2)-8 t(1) t(3) t(2)+3 t(3) t(2)-t(2)+t(1)^2 t(3)^2-t(1) t(3)^2-2 t(1)^2 t(3)+t(1) t(3)}{t(1) t(2)^{3/2} t(3)} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^5+4 q^4-9 q^3+16 q^2-20 q+24-23 q^{-1} +20 q^{-2} -14 q^{-3} +9 q^{-4} -3 q^{-5} + q^{-6} }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^6-3 z^2 a^4+a^4 z^{-2} +3 z^4 a^2+z^2 a^2-2 a^2 z^{-2} -3 a^2-z^6-z^4-2 z^2+ z^{-2} +1+2 z^4 a^{-2} +z^2 a^{-2} + a^{-2} -z^2 a^{-4} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^6 z^6-3 a^6 z^4+3 a^6 z^2-a^6+3 a^5 z^7-6 a^5 z^5+z^5 a^{-5} +3 a^5 z^3-z^3 a^{-5} +6 a^4 z^8-14 a^4 z^6+4 z^6 a^{-4} +15 a^4 z^4-5 z^4 a^{-4} -12 a^4 z^2+2 z^2 a^{-4} -a^4 z^{-2} +5 a^4+5 a^3 z^9-a^3 z^7+8 z^7 a^{-3} -18 a^3 z^5-11 z^5 a^{-3} +21 a^3 z^3+5 z^3 a^{-3} -9 a^3 z+2 a^3 z^{-1} +2 a^2 z^{10}+12 a^2 z^8+10 z^8 a^{-2} -34 a^2 z^6-14 z^6 a^{-2} +32 a^2 z^4+8 z^4 a^{-2} -23 a^2 z^2-2 z^2 a^{-2} -2 a^2 z^{-2} +11 a^2- a^{-2} +12 a z^9+7 z^9 a^{-1} -13 a z^7-z^7 a^{-1} -11 a z^5-11 z^5 a^{-1} +18 a z^3+6 z^3 a^{-1} -9 a z+2 a z^{-1} +2 z^{10}+16 z^8-37 z^6+27 z^4-12 z^2- z^{-2} +5 }[/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 11                       1 -1 9                     3   3 7                   6 1   -5 5                 10 3     7 3               11 7       -4 1             13 9         4 -1           12 13           1 -3         8 11             -3 -5       6 12               6 -7     3 8                 -5 -9     6                   6 -11 1 3                     -2 -13 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=1 }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/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.

L11a529 L11a531