L11a536

L11a535

L11a537

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Link Presentations

[edit Notes on L11a536'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 X12,22,13,21 X20,14,21,13 X14,20,9,19
Gauss code	{1, -2, 4, -5}, {2, -1, 6, -7}, {5, -4, 3, -9, 10, -11}, {7, -6, 8, -3, 11, -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{u v w^2 x-u v w^2+u v w x^2-3 u v w x+2 u v w-u v x^2+2 u v x-2 u w^2 x+u w^2-2 u w x^2+4 u w x-2 u w+2 u x^2-2 u x-2 v w^2 x+2 v w^2-2 v w x^2+4 v w x-2 v w+v x^2-2 v x+2 w^2 x-w^2+2 w x^2-3 w x+w-x^2+x}{\sqrt{u} \sqrt{v} w x} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{9/2}-4 q^{7/2}+7 q^{5/2}-12 q^{3/2}+14 \sqrt{q}-\frac{18}{\sqrt{q}}+\frac{15}{q^{3/2}}-\frac{15}{q^{5/2}}+\frac{9}{q^{7/2}}-\frac{6}{q^{9/2}}+\frac{2}{q^{11/2}}-\frac{1}{q^{13/2}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^7 z^{-1} +a^5 z^{-3} -3 a^5 z-a^5 z^{-1} +3 a^3 z^3-3 a^3 z^{-3} +z^3 a^{-3} -4 a^3 z^{-1} -a z^5-z^5 a^{-1} +2 a z^3+3 a z^{-3} -z^3 a^{-1} - a^{-1} z^{-3} +6 a z+7 a z^{-1} -3 z a^{-1} -3 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^7 z^5-3 a^7 z^3+3 a^7 z-a^7 z^{-1} +2 a^6 z^6-3 a^6 z^4+a^6+3 a^5 z^7-3 a^5 z^5+a^5 z^3-a^5 z^{-3} -3 a^5 z+3 a^5 z^{-1} +3 a^4 z^8+a^4 z^6+z^6 a^{-4} -10 a^4 z^4-2 z^4 a^{-4} +16 a^4 z^2+3 a^4 z^{-2} -11 a^4+2 a^3 z^9+6 a^3 z^7+4 z^7 a^{-3} -19 a^3 z^5-11 z^5 a^{-3} +26 a^3 z^3+7 z^3 a^{-3} -3 a^3 z^{-3} -21 a^3 z+12 a^3 z^{-1} +a^2 z^{10}+5 a^2 z^8+6 z^8 a^{-2} -3 a^2 z^6-16 z^6 a^{-2} -17 a^2 z^4+10 z^4 a^{-2} +33 a^2 z^2+6 a^2 z^{-2} -24 a^2+6 a z^9+4 z^9 a^{-1} -5 a z^7-4 z^7 a^{-1} -13 a z^5-9 z^5 a^{-1} +29 a z^3+14 z^3 a^{-1} -3 a z^{-3} - a^{-1} z^{-3} -28 a z-13 z a^{-1} +14 a z^{-1} +6 a^{-1} z^{-1} +z^{10}+8 z^8-19 z^6+2 z^4+17 z^2+3 z^{-2} -13 }[/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 10                       1 -1 8                     3   3 6                   4 1   -3 4                 8 3     5 2               9 7       -2 0             9 5         4 -2           7 10           3 -4         8 8             0 -6       2 8               6 -8     4 7                 -3 -10   1 5                   4 -12   1                     -1 -14 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{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\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^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\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^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\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^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/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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