L11a514

L11a513

L11a515

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Link Presentations

[edit Notes on L11a514's Link Presentations]

Planar diagram presentation	X8192 X16,5,17,6 X14,3,15,4 X20,12,21,11 X18,10,19,9 X4,15,5,16 X22,18,13,17 X12,20,7,19 X10,22,11,21 X2738 X6,13,1,14
Gauss code	{1, -10, 3, -6, 2, -11}, {10, -1, 5, -9, 4, -8}, {11, -3, 6, -2, 7, -5, 8, -4, 9, -7}

A Braid Representative	{{{braid_table}}}
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^2 v^2 w^2-u^2 v^2 w+u^2 v w^3-3 u^2 v w^2+2 u^2 v w-u^2 w^3+2 u^2 w^2-2 u v^2 w^2+3 u v^2 w-u v^2-2 u v w^3+5 u v w^2-5 u v w+2 u v+u w^3-3 u w^2+2 u w-2 v^2 w+v^2-2 v w^2+3 v w-v+w^2-w}{u v w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^4+3 q^3-5 q^2+10 q-12+15 q^{-1} -15 q^{-2} +14 q^{-3} -10 q^{-4} +7 q^{-5} -3 q^{-6} + q^{-7} }[/math] (db)
Signature	-2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^6 z^2+a^6-2 a^4 z^4-3 a^4 z^2+a^4 z^{-2} +a^4+a^2 z^6+a^2 z^4-z^4 a^{-2} -4 a^2 z^2-2 a^2 z^{-2} -2 z^2 a^{-2} -6 a^2+z^6+3 z^4+4 z^2+ z^{-2} +4 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^2 z^{10}+z^{10}+3 a^3 z^9+6 a z^9+3 z^9 a^{-1} +6 a^4 z^8+8 a^2 z^8+3 z^8 a^{-2} +5 z^8+7 a^5 z^7+6 a^3 z^7-11 a z^7-9 z^7 a^{-1} +z^7 a^{-3} +6 a^6 z^6-5 a^4 z^6-24 a^2 z^6-13 z^6 a^{-2} -26 z^6+3 a^7 z^5-8 a^5 z^5-25 a^3 z^5-5 a z^5+5 z^5 a^{-1} -4 z^5 a^{-3} +a^8 z^4-8 a^6 z^4-a^4 z^4+23 a^2 z^4+18 z^4 a^{-2} +33 z^4-2 a^7 z^3+4 a^5 z^3+22 a^3 z^3+14 a z^3+2 z^3 a^{-1} +4 z^3 a^{-3} -a^8 z^2+7 a^6 z^2-a^4 z^2-21 a^2 z^2-8 z^2 a^{-2} -20 z^2-9 a^3 z-9 a z-2 a^6+3 a^4+11 a^2+7+2 a^3 z^{-1} +2 a z^{-1} -a^4 z^{-2} -2 a^2 z^{-2} - 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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 9                       1 -1 7                     2   2 5                   3 1   -2 3                 7 2     5 1               6 4       -2 -1             9 6         3 -3           8 8           0 -5         6 7             -1 -7       4 8               4 -9     3 6                 -3 -11     4                   4 -13 1 3                     -2 -15 1                       1

Integral Khovanov Homology (db, data source)

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