L11a361

L11a360

L11a362

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Link Presentations

[edit Notes on L11a361's Link Presentations]

Planar diagram presentation	X12,1,13,2 X14,4,15,3 X22,14,11,13 X16,6,17,5 X2,11,3,12 X4,16,5,15 X6,22,7,21 X20,10,21,9 X18,8,19,7 X8,20,9,19 X10,18,1,17
Gauss code	{1, -5, 2, -6, 4, -7, 9, -10, 8, -11}, {5, -1, 3, -2, 6, -4, 11, -9, 10, -8, 7, -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{u^4 v^3-u^4 v^2+u^3 v^4-4 u^3 v^3+4 u^3 v^2-2 u^3 v-u^2 v^4+4 u^2 v^3-7 u^2 v^2+4 u^2 v-u^2-2 u v^3+4 u v^2-4 u v+u-v^2+v}{u^2 v^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -11 q^{9/2}+8 q^{7/2}-5 q^{5/2}+2 q^{3/2}+q^{23/2}-3 q^{21/2}+6 q^{19/2}-10 q^{17/2}+12 q^{15/2}-14 q^{13/2}+13 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} -4 z^5 a^{-5} -4 z^5 a^{-7} +z^5 a^{-9} +4 z^3 a^{-3} -4 z^3 a^{-5} -5 z^3 a^{-7} +3 z^3 a^{-9} +4 z a^{-3} -3 z a^{-7} +2 z a^{-9} + a^{-5} z^{-1} - a^{-7} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^{10} a^{-6} -z^{10} a^{-8} -2 z^9 a^{-5} -5 z^9 a^{-7} -3 z^9 a^{-9} -2 z^8 a^{-4} -z^8 a^{-6} -4 z^8 a^{-8} -5 z^8 a^{-10} -z^7 a^{-3} +5 z^7 a^{-5} +12 z^7 a^{-7} -6 z^7 a^{-11} +8 z^6 a^{-4} +8 z^6 a^{-6} +9 z^6 a^{-8} +4 z^6 a^{-10} -5 z^6 a^{-12} +5 z^5 a^{-3} -9 z^5 a^{-7} +6 z^5 a^{-9} +7 z^5 a^{-11} -3 z^5 a^{-13} -9 z^4 a^{-4} -4 z^4 a^{-6} +z^4 a^{-8} +2 z^4 a^{-10} +5 z^4 a^{-12} -z^4 a^{-14} -8 z^3 a^{-3} -3 z^3 a^{-5} +9 z^3 a^{-7} -3 z^3 a^{-9} -4 z^3 a^{-11} +3 z^3 a^{-13} +2 z^2 a^{-4} -2 z^2 a^{-8} -3 z^2 a^{-10} -2 z^2 a^{-12} +z^2 a^{-14} +4 z a^{-3} -2 z a^{-5} -5 z a^{-7} +2 z a^{-9} -z a^{-13} - a^{-6} + a^{-5} z^{-1} + a^{-7} 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                     2   2 20                   4 1   -3 18                 6 2     4 16               7 5       -2 14             7 5         2 12           6 7           1 10         5 7             -2 8       3 6               3 6     2 5                 -3 4   1 4                   3 2   1                     -1 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}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/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}^{5}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=7 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=8 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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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