L11a460

L11a459

L11a461

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Link Presentations

[edit Notes on L11a460's Link Presentations]

Planar diagram presentation	X6172 X14,4,15,3 X22,10,13,9 X20,8,21,7 X8,14,9,13 X18,15,19,16 X16,6,17,5 X12,18,5,17 X10,22,11,21 X2,11,3,12 X4,20,1,19
Gauss code	{1, -10, 2, -11}, {7, -1, 4, -5, 3, -9, 10, -8}, {5, -2, 6, -7, 8, -6, 11, -4, 9, -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{t(1) t(3)^2 t(2)^3-t(3)^2 t(2)^3-2 t(1) t(3) t(2)^3+2 t(3) t(2)^3-t(2)^3+t(1) t(3)^3 t(2)^2-t(3)^3 t(2)^2-5 t(1) t(3)^2 t(2)^2+5 t(3)^2 t(2)^2-2 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)+2 t(3)^3 t(2)+6 t(1) t(3)^2 t(2)-6 t(3)^2 t(2)+t(1) t(2)-5 t(1) t(3) t(2)+5 t(3) t(2)-t(2)+t(1) t(3)^3-2 t(1) t(3)^2+2 t(3)^2+t(1) t(3)-t(3)}{\sqrt{t(1)} t(2)^{3/2} t(3)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^9-3 q^8+8 q^7-12 q^6+19 q^5-22 q^4+23 q^3-20 q^2- q^{-2} +16 q+5 q^{-1} -10 }[/math] (db)
Signature	2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^6 a^{-2} +z^6 a^{-4} +z^4 a^{-2} +z^4 a^{-4} -2 z^4 a^{-6} -z^4+z^2 a^{-4} -3 z^2 a^{-6} +z^2 a^{-8} + a^{-4} -3 a^{-6} + a^{-8} +1+ a^{-4} z^{-2} -2 a^{-6} z^{-2} + a^{-8} z^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^6 a^{-10} -3 z^4 a^{-10} +2 z^2 a^{-10} +3 z^7 a^{-9} -7 z^5 a^{-9} +3 z^3 a^{-9} +6 z^8 a^{-8} -18 z^6 a^{-8} +24 z^4 a^{-8} -20 z^2 a^{-8} - a^{-8} z^{-2} +8 a^{-8} +5 z^9 a^{-7} -6 z^7 a^{-7} -7 z^5 a^{-7} +17 z^3 a^{-7} -11 z a^{-7} +2 a^{-7} z^{-1} +2 z^{10} a^{-6} +10 z^8 a^{-6} -36 z^6 a^{-6} +46 z^4 a^{-6} -31 z^2 a^{-6} -2 a^{-6} z^{-2} +13 a^{-6} +12 z^9 a^{-5} -19 z^7 a^{-5} +z^5 a^{-5} +17 z^3 a^{-5} -11 z a^{-5} +2 a^{-5} z^{-1} +2 z^{10} a^{-4} +15 z^8 a^{-4} -36 z^6 a^{-4} +26 z^4 a^{-4} -10 z^2 a^{-4} - a^{-4} z^{-2} +5 a^{-4} +7 z^9 a^{-3} -14 z^5 a^{-3} +6 z^3 a^{-3} +11 z^8 a^{-2} -14 z^6 a^{-2} +2 z^4 a^{-2} -z^2 a^{-2} +10 z^7 a^{-1} +a z^5-14 z^5 a^{-1} +3 z^3 a^{-1} +5 z^6-5 z^4+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   \ -3 -2 -1 0 1 2 3 4 5 6 7 8 χ 19                       1 1 17                     3 1 -2 15                   5     5 13                 7 3     -4 11               12 5       7 9             11 8         -3 7           12 11           1 5         8 11             3 3       8 12               -4 1     4 10                 6 -1   1 6                   -5 -3   4                     4 -5 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=3 }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=7 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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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