L11a347

L11a346

L11a348

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Link Presentations

[edit Notes on L11a347's Link Presentations]

Planar diagram presentation	X12,1,13,2 X8493 X20,14,21,13 X22,16,11,15 X14,22,15,21 X6,18,7,17 X18,8,19,7 X16,20,17,19 X10,6,1,5 X4,10,5,9 X2,11,3,12
Gauss code	{1, -11, 2, -10, 9, -6, 7, -2, 10, -9}, {11, -1, 3, -5, 4, -8, 6, -7, 8, -3, 5, -4}

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(2)-1) \left(2 t(2)^2 t(1)^2-t(2) t(1)^2+2 t(1)^2-t(2)^2 t(1)-t(1)+2 t(2)^2-t(2)+2\right)}{t(1)^{3/2} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -12 q^{9/2}+7 q^{7/2}-5 q^{5/2}+q^{3/2}+q^{23/2}-4 q^{21/2}+8 q^{19/2}-12 q^{17/2}+15 q^{15/2}-15 q^{13/2}+15 q^{11/2}-\sqrt{q} }[/math] (db)
Signature	5 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^7 a^{-5} -z^7 a^{-7} +z^5 a^{-3} -5 z^5 a^{-5} -3 z^5 a^{-7} +z^5 a^{-9} +5 z^3 a^{-3} -11 z^3 a^{-5} +2 z^3 a^{-9} +8 z a^{-3} -14 z a^{-5} +6 z a^{-7} +4 a^{-3} z^{-1} -8 a^{-5} z^{-1} +5 a^{-7} z^{-1} - a^{-9} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^4 a^{-14} +4 z^5 a^{-13} -2 z^3 a^{-13} +8 z^6 a^{-12} -8 z^4 a^{-12} +2 z^2 a^{-12} +10 z^7 a^{-11} -13 z^5 a^{-11} +4 z^3 a^{-11} +z a^{-11} +8 z^8 a^{-10} -9 z^6 a^{-10} +3 z^2 a^{-10} -2 a^{-10} +4 z^9 a^{-9} -5 z^5 a^{-9} -5 z^3 a^{-9} +3 z a^{-9} + a^{-9} z^{-1} +z^{10} a^{-8} +6 z^8 a^{-8} -10 z^6 a^{-8} -11 z^4 a^{-8} +20 z^2 a^{-8} -9 a^{-8} +5 z^9 a^{-7} -10 z^7 a^{-7} +3 z^5 a^{-7} +3 z^3 a^{-7} -7 z a^{-7} +5 a^{-7} z^{-1} +z^{10} a^{-6} -z^8 a^{-6} +5 z^6 a^{-6} -25 z^4 a^{-6} +33 z^2 a^{-6} -14 a^{-6} +z^9 a^{-5} +z^7 a^{-5} -15 z^5 a^{-5} +27 z^3 a^{-5} -21 z a^{-5} +8 a^{-5} z^{-1} +z^8 a^{-4} -2 z^6 a^{-4} -5 z^4 a^{-4} +14 z^2 a^{-4} -8 a^{-4} +z^7 a^{-3} -6 z^5 a^{-3} +13 z^3 a^{-3} -12 z a^{-3} +4 a^{-3} z^{-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   \ -2 -1 0 1 2 3 4 5 6 7 8 9 χ 24                       1 -1 22                     3   3 20                   5 1   -4 18                 7 3     4 16               8 5       -3 14             7 7         0 12           8 8           0 10         4 7             -3 8       3 8               5 6     2 4                 -2 4   1 5                   4 2                         0 0 1                       1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=4 }[/math] [math]\displaystyle{ i=6 }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=7 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=8 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=9 }[/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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