L11a542

L11a541

L11a543

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Link Presentations

[edit Notes on L11a542's Link Presentations]

Planar diagram presentation	X6172 X12,6,13,5 X8493 X2,16,3,15 X16,7,17,8 X22,20,15,19 X14,22,11,21 X20,14,21,13 X18,10,19,9 X10,12,5,11 X4,17,1,18
Gauss code	{1, -4, 3, -11}, {10, -2, 8, -7}, {2, -1, 5, -3, 9, -10}, {4, -5, 11, -9, 6, -8, 7, -6}

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{(u-1) (w-1) (x-1) (2 v w x+v (-w)-v x+v-w x+w+x-2)}{\sqrt{u} \sqrt{v} w x} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 25 q^{9/2}-27 q^{7/2}+20 q^{5/2}-17 q^{3/2}+\frac{1}{q^{3/2}}-q^{19/2}+5 q^{17/2}-11 q^{15/2}+17 q^{13/2}-24 q^{11/2}+8 \sqrt{q}-\frac{4}{\sqrt{q}} }[/math] (db)
Signature	3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^7 a^{-3} +z^7 a^{-5} -z^5 a^{-1} +3 z^5 a^{-3} +2 z^5 a^{-5} -z^5 a^{-7} -2 z^3 a^{-1} +4 z^3 a^{-3} -z^3 a^{-5} -z^3 a^{-7} +2 z a^{-3} -3 z a^{-5} +z a^{-7} +2 a^{-1} z^{-1} -4 a^{-3} z^{-1} +2 a^{-5} z^{-1} + a^{-1} z^{-3} -3 a^{-3} z^{-3} +3 a^{-5} z^{-3} - a^{-7} z^{-3} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^5 a^{-11} +5 z^6 a^{-10} -4 z^4 a^{-10} +11 z^7 a^{-9} -15 z^5 a^{-9} +6 z^3 a^{-9} +13 z^8 a^{-8} -16 z^6 a^{-8} +3 z^4 a^{-8} +z^2 a^{-8} +8 z^9 a^{-7} +6 z^7 a^{-7} -31 z^5 a^{-7} +20 z^3 a^{-7} + a^{-7} z^{-3} -3 z a^{-7} -2 a^{-7} z^{-1} +2 z^{10} a^{-6} +22 z^8 a^{-6} -43 z^6 a^{-6} +14 z^4 a^{-6} +3 z^2 a^{-6} -3 a^{-6} z^{-2} +4 a^{-6} +13 z^9 a^{-5} -5 z^7 a^{-5} -35 z^5 a^{-5} +36 z^3 a^{-5} +3 a^{-5} z^{-3} -11 z a^{-5} -3 a^{-5} z^{-1} +2 z^{10} a^{-4} +15 z^8 a^{-4} -32 z^6 a^{-4} +9 z^4 a^{-4} +4 z^2 a^{-4} -6 a^{-4} z^{-2} +7 a^{-4} +5 z^9 a^{-3} +4 z^7 a^{-3} -30 z^5 a^{-3} +32 z^3 a^{-3} +3 a^{-3} z^{-3} -11 z a^{-3} -3 a^{-3} z^{-1} +6 z^8 a^{-2} -9 z^6 a^{-2} +3 z^2 a^{-2} -3 a^{-2} z^{-2} +4 a^{-2} +4 z^7 a^{-1} -10 z^5 a^{-1} +10 z^3 a^{-1} + a^{-1} z^{-3} -3 z a^{-1} -2 a^{-1} z^{-1} +z^6-2 z^4+z^2 }[/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 20                       1 1 18                     4   -4 16                   7 1   6 14                 10 4     -6 12               14 7       7 10             13 12         -1 8           14 12           2 6         10 17             7 4       7 10               -3 2     3 12                 9 0   1 5                   -4 -2   3                     3 -4 1                       -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=2 }[/math] [math]\displaystyle{ i=4 }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{17}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{14} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{13} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{14} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=7 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=8 }[/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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