L11a161

L11a160

L11a162

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Link Presentations

[edit Notes on L11a161's Link Presentations]

Planar diagram presentation	X8192 X10,4,11,3 X22,10,7,9 X2738 X4,22,5,21 X14,5,15,6 X16,13,17,14 X20,18,21,17 X18,12,19,11 X12,20,13,19 X6,15,1,16
Gauss code	{1, -4, 2, -5, 6, -11}, {4, -1, 3, -2, 9, -10, 7, -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^3-13 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{ -8 q^{9/2}+\frac{1}{q^{9/2}}+14 q^{7/2}-\frac{4}{q^{7/2}}-19 q^{5/2}+\frac{8}{q^{5/2}}+21 q^{3/2}-\frac{14}{q^{3/2}}-q^{13/2}+4 q^{11/2}-22 \sqrt{q}+\frac{18}{\sqrt{q}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^3 a^{-5} -z a^{-5} +2 z^5 a^{-3} -a^3 z^3+5 z^3 a^{-3} -a^3 z+5 z a^{-3} + a^{-3} z^{-1} -z^7 a^{-1} +2 a z^5-4 z^5 a^{-1} +5 a z^3-9 z^3 a^{-1} +5 a z-9 z a^{-1} +2 a z^{-1} -3 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^{10} a^{-2} -z^{10}-4 a z^9-8 z^9 a^{-1} -4 z^9 a^{-3} -6 a^2 z^8-16 z^8 a^{-2} -7 z^8 a^{-4} -15 z^8-4 a^3 z^7-4 a z^7-4 z^7 a^{-1} -11 z^7 a^{-3} -7 z^7 a^{-5} -a^4 z^6+12 a^2 z^6+28 z^6 a^{-2} +3 z^6 a^{-4} -4 z^6 a^{-6} +34 z^6+10 a^3 z^5+28 a z^5+40 z^5 a^{-1} +33 z^5 a^{-3} +10 z^5 a^{-5} -z^5 a^{-7} +2 a^4 z^4-5 a^2 z^4-7 z^4 a^{-2} +9 z^4 a^{-4} +6 z^4 a^{-6} -17 z^4-8 a^3 z^3-28 a z^3-40 z^3 a^{-1} -27 z^3 a^{-3} -6 z^3 a^{-5} +z^3 a^{-7} -a^4 z^2-a^2 z^2-7 z^2 a^{-2} -8 z^2 a^{-4} -3 z^2 a^{-6} -2 z^2+2 a^3 z+11 a z+16 z a^{-1} +9 z a^{-3} +2 z a^{-5} +3 a^{-2} + a^{-4} +3-2 a z^{-1} -3 a^{-1} z^{-1} - a^{-3} z^{-1} }[/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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 14                       1 1 12                     3   -3 10                   5 1   4 8                 9 3     -6 6               10 5       5 4             11 9         -2 2           11 10           1 0         8 12             4 -2       6 10               -4 -4     3 9                 6 -6   1 5                   -4 -8   3                     3 -10 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=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=6 }[/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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