L11a195

L11a194

L11a196

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Link Presentations

[edit Notes on L11a195's Link Presentations]

Planar diagram presentation	X8192 X10,4,11,3 X22,10,7,9 X2738 X18,13,19,14 X6,12,1,11 X20,15,21,16 X14,19,15,20 X4,18,5,17 X16,6,17,5 X12,21,13,22
Gauss code	{1, -4, 2, -9, 10, -6}, {4, -1, 3, -2, 6, -11, 5, -8, 7, -10, 9, -5, 8, -7, 11, -3}

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^2 v^4-3 u^2 v^3+4 u^2 v^2-4 u^2 v+2 u^2-3 u v^4+8 u v^3-9 u v^2+8 u v-3 u+2 v^4-4 v^3+4 v^2-3 v+1}{u v^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{9/2}-3 q^{7/2}+6 q^{5/2}-12 q^{3/2}+15 \sqrt{q}-\frac{19}{\sqrt{q}}+\frac{19}{q^{3/2}}-\frac{17}{q^{5/2}}+\frac{13}{q^{7/2}}-\frac{8}{q^{9/2}}+\frac{4}{q^{11/2}}-\frac{1}{q^{13/2}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a z^7-2 a^3 z^5+4 a z^5-2 z^5 a^{-1} +a^5 z^3-5 a^3 z^3+8 a z^3-6 z^3 a^{-1} +z^3 a^{-3} +a^5 z-4 a^3 z+8 a z-6 z a^{-1} +2 z a^{-3} -a^3 z^{-1} +3 a z^{-1} -2 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^7 z^5-a^7 z^3+4 a^6 z^6-6 a^6 z^4+2 a^6 z^2+7 a^5 z^7-11 a^5 z^5+5 a^5 z^3-a^5 z+7 a^4 z^8-7 a^4 z^6+z^6 a^{-4} -2 a^4 z^4-3 z^4 a^{-4} +3 a^4 z^2+2 z^2 a^{-4} -a^4+4 a^3 z^9+5 a^3 z^7+3 z^7 a^{-3} -19 a^3 z^5-9 z^5 a^{-3} +14 a^3 z^3+9 z^3 a^{-3} -4 a^3 z-4 z a^{-3} +a^3 z^{-1} +a^2 z^{10}+12 a^2 z^8+4 z^8 a^{-2} -24 a^2 z^6-8 z^6 a^{-2} +12 a^2 z^4+2 z^4 a^{-2} +3 a^2 z^2+2 z^2 a^{-2} -3 a^2+7 a z^9+3 z^9 a^{-1} -4 a z^7+z^7 a^{-1} -15 a z^5-17 z^5 a^{-1} +21 a z^3+22 z^3 a^{-1} -11 a z-12 z a^{-1} +3 a z^{-1} +2 a^{-1} z^{-1} +z^{10}+9 z^8-22 z^6+13 z^4+2 z^2-3 }[/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 10                       1 -1 8                     2   2 6                   4 1   -3 4                 8 2     6 2               8 5       -3 0             11 7         4 -2           9 9           0 -4         8 10             -2 -6       5 9               4 -8     3 8                 -5 -10   1 5                   4 -12   3                     -3 -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}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\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^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\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^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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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