L11a159

L11a158

L11a160

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Link Presentations

[edit Notes on L11a159's Link Presentations]

Planar diagram presentation	X8192 X10,4,11,3 X22,10,7,9 X2738 X4,22,5,21 X14,5,15,6 X16,11,17,12 X20,17,21,18 X18,14,19,13 X12,20,13,19 X6,15,1,16
Gauss code	{1, -4, 2, -5, 6, -11}, {4, -1, 3, -2, 7, -10, 9, -6, 11, -7, 8, -9, 10, -8, 5, -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^4-4 u^2 v^3+6 u^2 v^2-4 u^2 v+u^2-2 u v^4+9 u v^3-15 u v^2+9 u v-2 u+v^4-4 v^3+6 v^2-4 v+1}{u v^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{9/2}-4 q^{7/2}+8 q^{5/2}-14 q^{3/2}+19 \sqrt{q}-\frac{22}{\sqrt{q}}+\frac{22}{q^{3/2}}-\frac{20}{q^{5/2}}+\frac{14}{q^{7/2}}-\frac{9}{q^{9/2}}+\frac{4}{q^{11/2}}-\frac{1}{q^{13/2}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^5 z^3+a^5 z+a^5 z^{-1} -2 a^3 z^5-5 a^3 z^3+z^3 a^{-3} -6 a^3 z+z a^{-3} -2 a^3 z^{-1} +a z^7+4 a z^5-2 z^5 a^{-1} +9 a z^3-5 z^3 a^{-1} +8 a z-5 z a^{-1} +2 a z^{-1} - a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^7 z^5-a^7 z^3+4 a^6 z^6-5 a^6 z^4+a^6 z^2+8 a^5 z^7-13 a^5 z^5+9 a^5 z^3-4 a^5 z+a^5 z^{-1} +9 a^4 z^8-12 a^4 z^6+z^6 a^{-4} +6 a^4 z^4-2 z^4 a^{-4} -2 a^4 z^2+z^2 a^{-4} +5 a^3 z^9+9 a^3 z^7+4 z^7 a^{-3} -36 a^3 z^5-10 z^5 a^{-3} +36 a^3 z^3+8 z^3 a^{-3} -14 a^3 z-2 z a^{-3} +2 a^3 z^{-1} +a^2 z^{10}+20 a^2 z^8+6 z^8 a^{-2} -46 a^2 z^6-12 z^6 a^{-2} +35 a^2 z^4+6 z^4 a^{-2} -10 a^2 z^2-z^2 a^{-2} +a^2+9 a z^9+4 z^9 a^{-1} +2 a z^7+5 z^7 a^{-1} -43 a z^5-31 z^5 a^{-1} +48 a z^3+30 z^3 a^{-1} -18 a z-10 z a^{-1} +2 a z^{-1} + a^{-1} z^{-1} +z^{10}+17 z^8-43 z^6+32 z^4-9 z^2 }[/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                   5 1   -4 4                 9 3     6 2               10 5       -5 0             12 9         3 -2           11 11           0 -4         9 11             -2 -6       6 12               6 -8     3 8                 -5 -10   1 6                   5 -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}^{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}^{9} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/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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