L11a423

L11a422

L11a424

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Link Presentations

[edit Notes on L11a423's Link Presentations]

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

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)^4+t(1) t(2) t(3)^4-2 t(2) t(3)^4+t(3)^4+t(1) t(2)^2 t(3)^3-3 t(2)^2 t(3)^3+3 t(1) t(3)^3-4 t(1) t(2) t(3)^3+6 t(2) t(3)^3-2 t(3)^3-2 t(1) t(2)^2 t(3)^2+4 t(2)^2 t(3)^2-4 t(1) t(3)^2+7 t(1) t(2) t(3)^2-7 t(2) t(3)^2+2 t(3)^2+2 t(1) t(2)^2 t(3)-3 t(2)^2 t(3)+3 t(1) t(3)-6 t(1) t(2) t(3)+4 t(2) t(3)-t(3)-t(1) t(2)^2+t(2)^2+2 t(1) t(2)-t(2)}{\sqrt{t(1)} t(2) t(3)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^8+4 q^7-10 q^6+16 q^5-21 q^4+25 q^3+ q^{-3} -23 q^2-3 q^{-2} +21 q+9 q^{-1} -14 }[/math] (db)
Signature	2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^4 a^{-6} -z^2 a^{-6} - a^{-6} z^{-2} -2 a^{-6} +z^6 a^{-4} +2 z^4 a^{-4} +6 z^2 a^{-4} +4 a^{-4} z^{-2} +9 a^{-4} +z^6 a^{-2} -z^4 a^{-2} +a^2 z^2-8 z^2 a^{-2} -5 a^{-2} z^{-2} +a^2-11 a^{-2} -2 z^4-z^2+2 z^{-2} +3 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^5 a^{-9} -z^3 a^{-9} +4 z^6 a^{-8} -4 z^4 a^{-8} +9 z^7 a^{-7} -14 z^5 a^{-7} +9 z^3 a^{-7} -4 z a^{-7} + a^{-7} z^{-1} +12 z^8 a^{-6} -22 z^6 a^{-6} +19 z^4 a^{-6} -9 z^2 a^{-6} - a^{-6} z^{-2} +3 a^{-6} +8 z^9 a^{-5} -2 z^7 a^{-5} -20 z^5 a^{-5} +29 z^3 a^{-5} -19 z a^{-5} +5 a^{-5} z^{-1} +2 z^{10} a^{-4} +20 z^8 a^{-4} -54 z^6 a^{-4} +52 z^4 a^{-4} -29 z^2 a^{-4} -4 a^{-4} z^{-2} +15 a^{-4} +13 z^9 a^{-3} -14 z^7 a^{-3} -21 z^5 a^{-3} +45 z^3 a^{-3} -33 z a^{-3} +9 a^{-3} z^{-1} +2 z^{10} a^{-2} +14 z^8 a^{-2} +a^2 z^6-43 z^6 a^{-2} -3 a^2 z^4+48 z^4 a^{-2} +3 a^2 z^2-38 z^2 a^{-2} -5 a^{-2} z^{-2} -a^2+20 a^{-2} +5 z^9 a^{-1} +3 a z^7-6 a z^5-22 z^5 a^{-1} +3 a z^3+29 z^3 a^{-1} -18 z a^{-1} +5 a^{-1} z^{-1} +6 z^8-14 z^6+16 z^4-15 z^2-2 z^{-2} +8 }[/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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 17                       1 -1 15                     3   3 13                   7 1   -6 11                 9 3     6 9               12 7       -5 7             13 9         4 5           12 14           2 3         9 11             -2 1       6 13               7 -1     3 8                 -5 -3     6                   6 -5 1 3                     -2 -7 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=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{13}\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^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=7 }[/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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