L11a467

L11a466

L11a468

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Link Presentations

[edit Notes on L11a467's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X18,12,19,11 X16,8,17,7 X8,16,9,15 X20,13,21,14 X22,20,15,19 X12,21,13,22 X14,18,5,17 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {5, -4, 9, -3, 7, -6, 8, -7}, {10, -1, 4, -5, 11, -2, 3, -8, 6, -9}

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{(t(3)-1) \left(-t(2) t(3)^3+t(3)^3+t(1) t(2)^2 t(3)^2-2 t(2)^2 t(3)^2+2 t(1) t(3)^2-3 t(1) t(2) t(3)^2+5 t(2) t(3)^2-2 t(3)^2-2 t(1) t(2)^2 t(3)+2 t(2)^2 t(3)-2 t(1) t(3)+5 t(1) t(2) t(3)-3 t(2) t(3)+t(3)+t(1) t(2)^2-t(1) t(2)\right)}{\sqrt{t(1)} t(2) t(3)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{-6} -q^5-3 q^{-5} +4 q^4+8 q^{-4} -9 q^3-13 q^{-3} +15 q^2+19 q^{-2} -19 q-21 q^{-1} +23 }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^6-3 a^4 z^2+a^4 z^{-2} -z^2 a^{-4} -a^4+3 a^2 z^4+2 z^4 a^{-2} +2 a^2 z^2-2 a^2 z^{-2} +z^2 a^{-2} -2 a^2-z^6-z^4-z^2+ z^{-2} +2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^6 z^6-3 a^6 z^4+3 a^6 z^2-a^6+3 a^5 z^7-7 a^5 z^5+z^5 a^{-5} +5 a^5 z^3-z^3 a^{-5} -a^5 z+5 a^4 z^8-10 a^4 z^6+4 z^6 a^{-4} +7 a^4 z^4-5 z^4 a^{-4} -5 a^4 z^2+2 z^2 a^{-4} -a^4 z^{-2} +4 a^4+4 a^3 z^9+a^3 z^7+8 z^7 a^{-3} -18 a^3 z^5-12 z^5 a^{-3} +20 a^3 z^3+7 z^3 a^{-3} -10 a^3 z-2 z a^{-3} +2 a^3 z^{-1} +a^2 z^{10}+16 a^2 z^8+9 z^8 a^{-2} -44 a^2 z^6-11 z^6 a^{-2} +46 a^2 z^4+5 z^4 a^{-2} -30 a^2 z^2-4 z^2 a^{-2} -2 a^2 z^{-2} +12 a^2+2 a^{-2} +9 a z^9+5 z^9 a^{-1} -a z^7+9 z^7 a^{-1} -31 a z^5-33 z^5 a^{-1} +34 a z^3+27 z^3 a^{-1} -14 a z-7 z a^{-1} +2 a z^{-1} +z^{10}+20 z^8-48 z^6+46 z^4-28 z^2- z^{-2} +10 }[/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 11                       1 -1 9                     3   3 7                   6 1   -5 5                 9 3     6 3               11 7       -4 1             12 8         4 -1           10 12           2 -3         9 11             -2 -5       5 11               6 -7     3 8                 -5 -9   1 6                   5 -11   2                     -2 -13 1                       1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-1 }[/math] [math]\displaystyle{ i=1 }[/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}^{6}\oplus{\mathbb Z}_2^{2} }[/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}^{11}\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^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/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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