L10a68

L10a67

L10a69

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Link Presentations

[edit Notes on L10a68's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X6718 X16,13,17,14 X14,6,15,5 X4,16,5,15 X20,18,7,17 X18,12,19,11 X12,20,13,19
Gauss code	{1, -2, 3, -7, 6, -4}, {4, -1, 2, -3, 9, -10, 5, -6, 7, -5, 8, -9, 10, -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^2 v^4-3 u^2 v^3+3 u^2 v^2-u^2 v-u v^4+3 u v^3-5 u v^2+3 u v-u-v^3+3 v^2-3 v+1}{u v^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{9/2}-\frac{3}{q^{9/2}}-2 q^{7/2}+\frac{5}{q^{7/2}}+4 q^{5/2}-\frac{8}{q^{5/2}}-7 q^{3/2}+\frac{9}{q^{3/2}}+\frac{1}{q^{11/2}}+8 \sqrt{q}-\frac{10}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^3 z^5-3 a^3 z^3+z^3 a^{-3} -2 a^3 z+3 z a^{-3} + a^{-3} z^{-1} +a z^7+5 a z^5-2 z^5 a^{-1} +9 a z^3-8 z^3 a^{-1} +7 a z-9 z a^{-1} +2 a z^{-1} -3 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -a z^9-z^9 a^{-1} -3 a^2 z^8-2 z^8 a^{-2} -5 z^8-4 a^3 z^7-4 a z^7-2 z^7 a^{-1} -2 z^7 a^{-3} -4 a^4 z^6+3 a^2 z^6+4 z^6 a^{-2} -z^6 a^{-4} +12 z^6-3 a^5 z^5+4 a^3 z^5+14 a z^5+14 z^5 a^{-1} +7 z^5 a^{-3} -a^6 z^4+4 a^4 z^4-a^2 z^4+3 z^4 a^{-2} +4 z^4 a^{-4} -7 z^4+4 a^5 z^3-a^3 z^3-17 a z^3-19 z^3 a^{-1} -7 z^3 a^{-3} +a^6 z^2-a^2 z^2-7 z^2 a^{-2} -4 z^2 a^{-4} -3 z^2-a^5 z+9 a z+12 z a^{-1} +4 z a^{-3} +3 a^{-2} + a^{-4} +3-2 a z^{-1} -3 a^{-1} z^{-1} - a^{-3} z^{-1} }[/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 5 χ 10                     1 -1 8                   1   1 6                 3 1   -2 4               4 1     3 2             4 3       -1 0           6 4         2 -2         4 5           1 -4       4 5             -1 -6     2 5               3 -8   1 3                 -2 -10   2                   2 -12 1                     -1

Integral Khovanov Homology (db, data source)

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