L11a426

L11a425

L11a427

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Link Presentations

[edit Notes on L11a426's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X14,5,15,6 X20,11,21,12 X22,17,11,18 X16,21,17,22 X10,13,5,14 X8,20,9,19 X18,8,19,7 X2,9,3,10 X4,16,1,15
Gauss code	{1, -10, 2, -11}, {3, -1, 9, -8, 10, -7}, {4, -2, 7, -3, 11, -6, 5, -9, 8, -4, 6, -5}

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^2 w^4-3 u v^2 w^3+4 u v^2 w^2-2 u v^2 w-u v w^4+5 u v w^3-8 u v w^2+6 u v w-u v-u w^3+3 u w^2-3 u w+u-v^2 w^4+3 v^2 w^3-3 v^2 w^2+v^2 w+v w^4-6 v w^3+8 v w^2-5 v w+v+2 w^3-4 w^2+3 w-1}{\sqrt{u} v w^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ - q^{-8} +4 q^{-7} -10 q^{-6} +16 q^{-5} -22 q^{-4} +q^3+26 q^{-3} -4 q^2-24 q^{-2} +10 q+23 q^{-1} -15 }[/math] (db)
Signature	-2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^2 z^8+2 a^4 z^6-5 a^2 z^6+z^6-a^6 z^4+7 a^4 z^4-11 a^2 z^4+3 z^4-2 a^6 z^2+10 a^4 z^2-13 a^2 z^2+4 z^2-2 a^6+8 a^4-10 a^2+4-a^6 z^{-2} +4 a^4 z^{-2} -5 a^2 z^{-2} +2 z^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^9 z^5-a^9 z^3+4 a^8 z^6-4 a^8 z^4+a^8 z^2+9 a^7 z^7-14 a^7 z^5+11 a^7 z^3-5 a^7 z+a^7 z^{-1} +11 a^6 z^8-15 a^6 z^6+9 a^6 z^4-5 a^6 z^2-a^6 z^{-2} +3 a^6+7 a^5 z^9+7 a^5 z^7-37 a^5 z^5+38 a^5 z^3-18 a^5 z+5 a^5 z^{-1} +2 a^4 z^{10}+22 a^4 z^8-56 a^4 z^6+51 a^4 z^4-28 a^4 z^2-4 a^4 z^{-2} +12 a^4+14 a^3 z^9-13 a^3 z^7-23 a^3 z^5+36 a^3 z^3-24 a^3 z+9 a^3 z^{-1} +2 a^2 z^{10}+19 a^2 z^8-57 a^2 z^6+z^6 a^{-2} +58 a^2 z^4-2 z^4 a^{-2} -35 a^2 z^2+z^2 a^{-2} -5 a^2 z^{-2} +15 a^2+7 a z^9-7 a z^7+4 z^7 a^{-1} -9 a z^5-8 z^5 a^{-1} +14 a z^3+4 z^3 a^{-1} -11 a z+5 a z^{-1} +8 z^8-19 z^6+18 z^4-12 z^2-2 z^{-2} +7 }[/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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 7                       1 1 5                     3   -3 3                   7 1   6 1                 9 4     -5 -1               14 6       8 -3             12 11         -1 -5           14 12           2 -7         9 13             4 -9       7 13               -6 -11     3 9                 6 -13   1 7                   -6 -15   3                     3 -17 1                       -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-3 }[/math] [math]\displaystyle{ i=-1 }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{13} }[/math] [math]\displaystyle{ {\mathbb Z}^{14} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{14} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=4 }[/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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