L11n410

L11n409

L11n411

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Link Presentations

[edit Notes on L11n410's Link Presentations]

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

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-1) (w-1) \left(2 v w^2-2 v w+v-w^2+2 w-2\right)}{\sqrt{u} \sqrt{v} w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 1-3 q^{-1} +7 q^{-2} -9 q^{-3} +14 q^{-4} -13 q^{-5} +13 q^{-6} -10 q^{-7} +7 q^{-8} -3 q^{-9} }[/math] (db)
Signature	-4 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^{10} z^{-2} -a^{10}+a^8 z^4+3 a^8 z^2+4 a^8 z^{-2} +6 a^8-a^6 z^6-3 a^6 z^4-6 a^6 z^2-5 a^6 z^{-2} -10 a^6-a^4 z^6-2 a^4 z^4+a^4 z^2+2 a^4 z^{-2} +4 a^4+a^2 z^4+2 a^2 z^2+a^2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ 6 z^3 a^{11}-5 z a^{11}+a^{11} z^{-1} +3 z^6 a^{10}+3 z^4 a^{10}-6 z^2 a^{10}-a^{10} z^{-2} +3 a^{10}+8 z^7 a^9-19 z^5 a^9+29 z^3 a^9-20 z a^9+5 a^9 z^{-1} +7 z^8 a^8-16 z^6 a^8+25 z^4 a^8-23 z^2 a^8-4 a^8 z^{-2} +13 a^8+2 z^9 a^7+9 z^7 a^7-33 z^5 a^7+41 z^3 a^7-30 z a^7+9 a^7 z^{-1} +11 z^8 a^6-29 z^6 a^6+30 z^4 a^6-25 z^2 a^6-5 a^6 z^{-2} +16 a^6+2 z^9 a^5+4 z^7 a^5-22 z^5 a^5+23 z^3 a^5-15 z a^5+5 a^5 z^{-1} +4 z^8 a^4-9 z^6 a^4+5 z^4 a^4-5 z^2 a^4-2 a^4 z^{-2} +6 a^4+3 z^7 a^3-8 z^5 a^3+5 z^3 a^3+z^6 a^2-3 z^4 a^2+3 z^2 a^2-a^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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 1                   1 1 -1                 2   -2 -3               5 1   4 -5             6 4     -2 -7           8 3       5 -9         5 6         1 -11       8 8           0 -13     4 7             3 -15   3 6               -3 -17   4                 4 -19 3                   -3

Integral Khovanov Homology (db, data source)

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