L11n295

L11n294

L11n296

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Link Presentations

[edit Notes on L11n295's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X20,16,21,15 X14,8,15,7 X21,10,22,5 X11,19,12,18 X9,17,10,16 X17,11,18,22 X8,19,9,20 X2536 X4,14,1,13
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -9, -7, 5}, {-6, -2, 11, -4, 3, 7, -8, 6, 9, -3, -5, 8}

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 v^2 w^2-u v^2 w-3 u v w^2+4 u v w-2 u v+u w^2-2 u w+u-v^2 w^2+2 v^2 w-v^2+2 v w^2-4 v w+3 v+w-1}{\sqrt{u} v w} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^8+3 q^7-6 q^6+8 q^5-10 q^4+11 q^3-8 q^2+8 q-3+2 q^{-1} }[/math] (db)
Signature	2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^4 a^{-6} -2 z^2 a^{-6} - a^{-6} z^{-2} -2 a^{-6} +z^6 a^{-4} +4 z^4 a^{-4} +8 z^2 a^{-4} +4 a^{-4} z^{-2} +8 a^{-4} -3 z^4 a^{-2} -9 z^2 a^{-2} -5 a^{-2} z^{-2} -10 a^{-2} +2 z^2+2 z^{-2} +4 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^5 a^{-9} -2 z^3 a^{-9} +3 z^6 a^{-8} -6 z^4 a^{-8} +z^2 a^{-8} +5 z^7 a^{-7} -13 z^5 a^{-7} +11 z^3 a^{-7} -5 z a^{-7} + a^{-7} z^{-1} +4 z^8 a^{-6} -9 z^6 a^{-6} +9 z^4 a^{-6} -6 z^2 a^{-6} - a^{-6} z^{-2} +3 a^{-6} +z^9 a^{-5} +6 z^7 a^{-5} -23 z^5 a^{-5} +32 z^3 a^{-5} -18 z a^{-5} +5 a^{-5} z^{-1} +6 z^8 a^{-4} -18 z^6 a^{-4} +30 z^4 a^{-4} -24 z^2 a^{-4} -4 a^{-4} z^{-2} +12 a^{-4} +z^9 a^{-3} +2 z^7 a^{-3} -10 z^5 a^{-3} +24 z^3 a^{-3} -24 z a^{-3} +9 a^{-3} z^{-1} +2 z^8 a^{-2} -6 z^6 a^{-2} +18 z^4 a^{-2} -25 z^2 a^{-2} -5 a^{-2} z^{-2} +15 a^{-2} +z^7 a^{-1} -z^5 a^{-1} +5 z^3 a^{-1} -11 z a^{-1} +5 a^{-1} z^{-1} +3 z^4-8 z^2-2 z^{-2} +7 }[/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               4 1   -3 11             4 2     2 9           6 4       -2 7         5 4         1 5       4 7           3 3     4 4             0 1   1 6               5 -1 1 2                 -1 -3 2                   2

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