L11n364

L11n363

L11n365

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Link Presentations

[edit Notes on L11n364's Link Presentations]

Planar diagram presentation	X6172 X10,4,11,3 X12,8,13,7 X15,17,16,22 X9,18,10,19 X17,8,18,9 X13,21,14,20 X21,15,22,14 X19,5,20,16 X2536 X4,12,1,11
Gauss code	{1, -10, 2, -11}, {-6, 5, -9, 7, -8, 4}, {10, -1, 3, 6, -5, -2, 11, -3, -7, 8, -4, 9}

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) \left(v^2 w-2 v w+2 v-1\right)}{\sqrt{u} v w} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^8+3 q^7-5 q^6+7 q^5-8 q^4+9 q^3-6 q^2+6 q+ q^{-1} -2 }[/math] (db)
Signature	2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^4 a^{-6} -2 z^2 a^{-6} - a^{-6} +z^6 a^{-4} +4 z^4 a^{-4} +6 z^2 a^{-4} + a^{-4} z^{-2} +3 a^{-4} -2 z^4 a^{-2} -5 z^2 a^{-2} -2 a^{-2} z^{-2} -4 a^{-2} +z^2+ z^{-2} +2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^5 a^{-9} -2 z^3 a^{-9} +3 z^6 a^{-8} -7 z^4 a^{-8} +2 z^2 a^{-8} +4 z^7 a^{-7} -10 z^5 a^{-7} +6 z^3 a^{-7} -z a^{-7} +3 z^8 a^{-6} -7 z^6 a^{-6} +7 z^4 a^{-6} -5 z^2 a^{-6} +2 a^{-6} +z^9 a^{-5} +z^7 a^{-5} -6 z^5 a^{-5} +9 z^3 a^{-5} -3 z a^{-5} +4 z^8 a^{-4} -14 z^6 a^{-4} +27 z^4 a^{-4} -21 z^2 a^{-4} - a^{-4} z^{-2} +7 a^{-4} +z^9 a^{-3} -3 z^7 a^{-3} +7 z^5 a^{-3} -z^3 a^{-3} -4 z a^{-3} +2 a^{-3} z^{-1} +z^8 a^{-2} -4 z^6 a^{-2} +14 z^4 a^{-2} -17 z^2 a^{-2} -2 a^{-2} z^{-2} +7 a^{-2} +2 z^5 a^{-1} -2 z^3 a^{-1} -2 z a^{-1} +2 a^{-1} z^{-1} +z^4-3 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   \ -2 -1 0 1 2 3 4 5 6 7 χ 17                   1 -1 15                 2   2 13               3 1   -2 11             4 2     2 9           4 3       -1 7         5 4         1 5       3 6           3 3     3 3             0 1   1 5               4 -1   1                 -1 -3 1                   1

Integral Khovanov Homology (db, data source)

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