L11n258

L11n257

L11n259

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Link Presentations

[edit Notes on L11n258's Link Presentations]

Planar diagram presentation	X6172 X3,11,4,10 X7,15,8,14 X13,5,14,8 X11,18,12,19 X19,22,20,9 X15,20,16,21 X21,16,22,17 X17,12,18,13 X2536 X9,1,10,4
Gauss code	{1, -10, -2, 11}, {10, -1, -3, 4}, {-11, 2, -5, 9, -4, 3, -7, 8, -9, 5, -6, 7, -8, 6}

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 v w^3-4 u v w^2+5 u v w-3 u v-u w^3+2 u w^2-2 u w+u-v w^3+2 v w^2-2 v w+v+3 w^3-5 w^2+4 w-1}{\sqrt{u} \sqrt{v} w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -3 q^{-6} +6 q^{-5} -10 q^{-4} +q^3+13 q^{-3} -2 q^2-12 q^{-2} +7 q+13 q^{-1} -9 }[/math] (db)
Signature	-2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^6 z^{-2} -2 a^6-z^4 a^4+2 z^2 a^4+3 a^4 z^{-2} +6 a^4+z^6 a^2+2 z^4 a^2-2 a^2 z^{-2} -3 a^2-2 z^4-4 z^2- z^{-2} -3+z^2 a^{-2} + a^{-2} z^{-2} +2 a^{-2} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ 6 a^7 z^3-7 a^7 z+2 a^7 z^{-1} +3 a^6 z^6+a^6 z^2-a^6 z^{-2} +a^6+7 a^5 z^7-17 a^5 z^5+32 a^5 z^3-27 a^5 z+8 a^5 z^{-1} +5 a^4 z^8-4 a^4 z^6-2 a^4 z^4+a^4 z^2-3 a^4 z^{-2} +5 a^4+a^3 z^9+14 a^3 z^7-43 a^3 z^5+51 a^3 z^3-34 a^3 z+10 a^3 z^{-1} +8 a^2 z^8-13 a^2 z^6+z^6 a^{-2} +a^2 z^4-4 z^4 a^{-2} -2 a^2 z^2+6 z^2 a^{-2} -2 a^2 z^{-2} + a^{-2} z^{-2} +4 a^2-4 a^{-2} +a z^9+9 a z^7+2 z^7 a^{-1} -30 a z^5-4 z^5 a^{-1} +25 a z^3-10 a z+4 z a^{-1} +2 a z^{-1} -2 a^{-1} z^{-1} +3 z^8-5 z^6-z^4+4 z^2+ z^{-2} -3 }[/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   \ -5 -4 -3 -2 -1 0 1 2 3 4 χ 7                   1 1 5                 2 1 -1 3               5     5 1             4 2     -2 -1           9 5       4 -3         6 7         1 -5       7 6           1 -7     3 6             3 -9   3 7               -4 -11   3                 3 -13 3                   -3

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=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\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^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/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=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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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