L11a457

L11a456

L11a458

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Link Presentations

[edit Notes on L11a457's Link Presentations]

Planar diagram presentation	X6172 X14,3,15,4 X20,12,21,11 X18,8,19,7 X22,9,13,10 X16,21,17,22 X8,18,9,17 X10,15,11,16 X12,20,5,19 X2536 X4,13,1,14
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -7, 5, -8, 3, -9}, {11, -2, 8, -6, 7, -4, 9, -3, 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{-t(3)^2 t(2)^3+t(1) t(2)^3-t(1) t(3) t(2)^3+2 t(3) t(2)^3-t(2)^3+t(1) t(3)^3 t(2)^2-2 t(3)^3 t(2)^2-4 t(1) t(3)^2 t(2)^2+5 t(3)^2 t(2)^2-3 t(1) t(2)^2+6 t(1) t(3) t(2)^2-6 t(3) t(2)^2+2 t(2)^2-2 t(1) t(3)^3 t(2)+3 t(3)^3 t(2)+6 t(1) t(3)^2 t(2)-6 t(3)^2 t(2)+2 t(1) t(2)-5 t(1) t(3) t(2)+4 t(3) t(2)-t(2)+t(1) t(3)^3-t(3)^3-2 t(1) t(3)^2+t(3)^2+t(1) t(3)}{\sqrt{t(1)} t(2)^{3/2} t(3)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^3-4 q^2+10 q-15+21 q^{-1} -22 q^{-2} +23 q^{-3} -19 q^{-4} +14 q^{-5} -7 q^{-6} +3 q^{-7} - q^{-8} }[/math] (db)
Signature	-2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^8+3 a^6 z^2+2 a^6-3 a^4 z^4-2 a^4 z^2+a^4 z^{-2} +a^4+a^2 z^6-3 a^2 z^2-2 a^2 z^{-2} +z^2 a^{-2} -5 a^2-2 z^4+ z^{-2} +3 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^9 z^5-2 a^9 z^3+a^9 z+3 a^8 z^6-5 a^8 z^4+3 a^8 z^2-a^8+5 a^7 z^7-5 a^7 z^5+a^7 z+7 a^6 z^8-9 a^6 z^6+6 a^6 z^4+6 a^5 z^9-3 a^5 z^7-6 a^5 z^5+10 a^5 z^3-3 a^5 z+2 a^4 z^{10}+15 a^4 z^8-45 a^4 z^6+52 a^4 z^4-30 a^4 z^2-a^4 z^{-2} +9 a^4+13 a^3 z^9-21 a^3 z^7+3 a^3 z^5+10 a^3 z^3-8 a^3 z+2 a^3 z^{-1} +2 a^2 z^{10}+16 a^2 z^8-53 a^2 z^6+z^6 a^{-2} +59 a^2 z^4-2 z^4 a^{-2} -39 a^2 z^2+z^2 a^{-2} -2 a^2 z^{-2} +13 a^2+7 a z^9-9 a z^7+4 z^7 a^{-1} -5 a z^5-8 z^5 a^{-1} +6 a z^3+4 z^3 a^{-1} -5 a z+2 a z^{-1} +8 z^8-19 z^6+16 z^4-11 z^2- z^{-2} +6 }[/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                 8 3     -5 -1               13 7       6 -3             12 11         -1 -5           11 10           1 -7         8 12             4 -9       6 11               -5 -11     2 9                 7 -13   1 5                   -4 -15   2                     2 -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}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\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^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7} }[/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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