L11a476

L11a475

L11a477

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Link Presentations

[edit Notes on L11a476's Link Presentations]

Planar diagram presentation	X6172 X10,4,11,3 X14,8,15,7 X22,16,17,15 X16,18,5,17 X18,9,19,10 X20,13,21,14 X12,19,13,20 X8,21,9,22 X2536 X4,12,1,11
Gauss code	{1, -10, 2, -11}, {5, -6, 8, -7, 9, -4}, {10, -1, 3, -9, 6, -2, 11, -8, 7, -3, 4, -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(1)-1) (t(3)-1)^2 (t(3) t(2)-t(2)+1) (t(2) t(3)-t(3)+1)}{\sqrt{t(1)} t(2) t(3)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^5+4 q^4-10 q^3+16 q^2-20 q+25-22 q^{-1} +20 q^{-2} -14 q^{-3} +8 q^{-4} -3 q^{-5} + q^{-6} }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^8-2 a^2 z^6-z^6 a^{-2} +5 z^6+a^4 z^4-8 a^2 z^4-3 z^4 a^{-2} +11 z^4+3 a^4 z^2-13 a^2 z^2-4 z^2 a^{-2} +14 z^2+3 a^4-10 a^2-3 a^{-2} +10+a^4 z^{-2} -2 a^2 z^{-2} + z^{-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^8-9 a^4 z^6+4 z^6 a^{-4} +4 a^4 z^4-4 z^4 a^{-4} -2 a^4 z^2-a^4 z^{-2} +3 a^4+5 a^3 z^9-4 a^3 z^7+9 z^7 a^{-3} -7 a^3 z^5-14 z^5 a^{-3} +9 a^3 z^3+8 z^3 a^{-3} -7 a^3 z-3 z a^{-3} +2 a^3 z^{-1} +2 a^2 z^{10}+12 a^2 z^8+12 z^8 a^{-2} -38 a^2 z^6-23 z^6 a^{-2} +43 a^2 z^4+21 z^4 a^{-2} -31 a^2 z^2-12 z^2 a^{-2} -2 a^2 z^{-2} +13 a^2+5 a^{-2} +13 a z^9+8 z^9 a^{-1} -20 a z^7-4 z^7 a^{-1} +a z^5-14 z^5 a^{-1} +15 a z^3+20 z^3 a^{-1} -13 a z-10 z a^{-1} +2 a z^{-1} +2 z^{10}+19 z^8-55 z^6+61 z^4-38 z^2- z^{-2} +15 }[/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                   7 1   -6 5                 9 3     6 3               11 7       -4 1             14 9         5 -1           10 13           3 -3         10 12             -2 -5       6 12               6 -7     2 8                 -6 -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}^{2} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{14} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\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^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/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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