L10a41

L10a40

L10a42

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Link Presentations

[edit Notes on L10a41's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X16,10,17,9 X14,12,15,11 X10,16,11,15 X20,17,5,18 X18,7,19,8 X8,19,9,20 X2536 X4,14,1,13
Gauss code	{1, -9, 2, -10}, {9, -1, 7, -8, 3, -5, 4, -2, 10, -4, 5, -3, 6, -7, 8, -6}

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{(t(1)-1) (t(2)-1) \left(2 t(2)^2-3 t(2)+2\right)}{\sqrt{t(1)} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{9/2}-\frac{2}{q^{9/2}}-3 q^{7/2}+\frac{4}{q^{7/2}}+5 q^{5/2}-\frac{7}{q^{5/2}}-7 q^{3/2}+\frac{8}{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{ -z a^5-a^5 z^{-1} +2 z^3 a^3+4 z a^3+3 a^3 z^{-1} -z^5 a-2 z^3 a-3 z a-2 a z^{-1} -z^5 a^{-1} -2 z^3 a^{-1} -z a^{-1} +z^3 a^{-3} +z a^{-3} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^6 z^4-2 a^6 z^2+a^6+2 a^5 z^5-3 a^5 z^3+2 a^5 z-a^5 z^{-1} +2 a^4 z^6+z^6 a^{-4} +a^4 z^4-3 z^4 a^{-4} -5 a^4 z^2+2 z^2 a^{-4} +3 a^4+2 a^3 z^7+3 z^7 a^{-3} +2 a^3 z^5-10 z^5 a^{-3} -7 a^3 z^3+8 z^3 a^{-3} +7 a^3 z-2 z a^{-3} -3 a^3 z^{-1} +2 a^2 z^8+3 z^8 a^{-2} -a^2 z^6-8 z^6 a^{-2} +2 a^2 z^4+3 z^4 a^{-2} -4 a^2 z^2+z^2 a^{-2} +3 a^2+a z^9+z^9 a^{-1} +2 a z^7+3 z^7 a^{-1} -5 a z^5-15 z^5 a^{-1} +12 z^3 a^{-1} +5 a z-2 z a^{-1} -2 a z^{-1} +5 z^8-12 z^6+8 z^4-2 z^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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 10                     1 -1 8                   2   2 6                 3 1   -2 4               4 2     2 2             4 3       -1 0           6 4         2 -2         4 6           2 -4       3 4             -1 -6     1 4               3 -8   1 3                 -2 -10   1                   1 -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}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} }[/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}^{2}\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^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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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