L11a158

L11a157

L11a159

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Link Presentations

[edit Notes on L11a158's Link Presentations]

Planar diagram presentation	X8192 X10,4,11,3 X22,10,7,9 X2738 X16,12,17,11 X14,5,15,6 X4,15,5,16 X20,18,21,17 X18,13,19,14 X12,19,13,20 X6,22,1,21
Gauss code	{1, -4, 2, -7, 6, -11}, {4, -1, 3, -2, 5, -10, 9, -6, 7, -5, 8, -9, 10, -8, 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^4-4 u^2 v^3+6 u^2 v^2-4 u^2 v+u^2-2 u v^4+7 u v^3-9 u v^2+7 u v-2 u+v^4-4 v^3+6 v^2-4 v+1}{u v^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -4 q^{9/2}+\frac{3}{q^{9/2}}+8 q^{7/2}-\frac{7}{q^{7/2}}-13 q^{5/2}+\frac{11}{q^{5/2}}+17 q^{3/2}-\frac{16}{q^{3/2}}+q^{11/2}-\frac{1}{q^{11/2}}-19 \sqrt{q}+\frac{18}{\sqrt{q}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^7 a^{-1} +3 a z^5-4 z^5 a^{-1} +z^5 a^{-3} -3 a^3 z^3+9 a z^3-7 z^3 a^{-1} +2 z^3 a^{-3} +a^5 z-6 a^3 z+8 a z-5 z a^{-1} +z a^{-3} +a^5 z^{-1} -2 a^3 z^{-1} +2 a z^{-1} - a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^4 a^{-6} +a^5 z^7-4 a^5 z^5+4 z^5 a^{-5} +6 a^5 z^3-2 z^3 a^{-5} -4 a^5 z+a^5 z^{-1} +3 a^4 z^8-11 a^4 z^6+8 z^6 a^{-4} +13 a^4 z^4-7 z^4 a^{-4} -5 a^4 z^2+2 z^2 a^{-4} +3 a^3 z^9-4 a^3 z^7+11 z^7 a^{-3} -13 a^3 z^5-14 z^5 a^{-3} +26 a^3 z^3+7 z^3 a^{-3} -14 a^3 z-2 z a^{-3} +2 a^3 z^{-1} +a^2 z^{10}+10 a^2 z^8+10 z^8 a^{-2} -40 a^2 z^6-11 z^6 a^{-2} +40 a^2 z^4-z^4 a^{-2} -12 a^2 z^2+2 z^2 a^{-2} +a^2+8 a z^9+5 z^9 a^{-1} -8 a z^7+8 z^7 a^{-1} -30 a z^5-39 z^5 a^{-1} +44 a z^3+33 z^3 a^{-1} -18 a z-10 z a^{-1} +2 a z^{-1} + a^{-1} z^{-1} +z^{10}+17 z^8-48 z^6+34 z^4-7 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 12                       1 -1 10                     3   3 8                   5 1   -4 6                 8 3     5 4               9 5       -4 2             10 8         2 0           9 10           1 -2         7 9             -2 -4       5 10               5 -6     2 6                 -4 -8   1 5                   4 -10   2                     -2 -12 1                       1

Integral Khovanov Homology (db, data source)

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