L10a70

L10a69

L10a71

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Link Presentations

[edit Notes on L10a70's Link Presentations]

Planar diagram presentation	X8192 X18,9,19,10 X14,6,15,5 X16,12,17,11 X10,4,11,3 X12,7,13,8 X20,15,7,16 X6,14,1,13 X4,19,5,20 X2,18,3,17
Gauss code	{1, -10, 5, -9, 3, -8}, {6, -1, 2, -5, 4, -6, 8, -3, 7, -4, 10, -2, 9, -7}

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)^2 t(2)^4-2 t(1) t(2)^4+t(2)^4-3 t(1)^2 t(2)^3+7 t(1) t(2)^3-4 t(2)^3+5 t(1)^2 t(2)^2-9 t(1) t(2)^2+5 t(2)^2-4 t(1)^2 t(2)+7 t(1) t(2)-3 t(2)+t(1)^2-2 t(1)+1}{t(1) t(2)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{9/2}-\frac{4}{q^{9/2}}-5 q^{7/2}+\frac{9}{q^{7/2}}+9 q^{5/2}-\frac{14}{q^{5/2}}-14 q^{3/2}+\frac{17}{q^{3/2}}+\frac{1}{q^{11/2}}+17 \sqrt{q}-\frac{19}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^3 z^5-2 a^3 z^3+z^3 a^{-3} -2 a^3 z- a^{-3} z^{-1} +a z^7+4 a z^5-2 z^5 a^{-1} +7 a z^3-4 z^3 a^{-1} +4 a z-z a^{-1} + a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^6 z^4+4 a^5 z^5-a^5 z^3+9 a^4 z^6+z^6 a^{-4} -8 a^4 z^4-z^4 a^{-4} +3 a^4 z^2+13 a^3 z^7+5 z^7 a^{-3} -19 a^3 z^5-11 z^5 a^{-3} +12 a^3 z^3+5 z^3 a^{-3} -3 a^3 z+2 z a^{-3} - a^{-3} z^{-1} +11 a^2 z^8+8 z^8 a^{-2} -13 a^2 z^6-19 z^6 a^{-2} +a^2 z^4+11 z^4 a^{-2} +a^2 z^2-z^2 a^{-2} + a^{-2} +4 a z^9+4 z^9 a^{-1} +13 a z^7+5 z^7 a^{-1} -42 a z^5-30 z^5 a^{-1} +27 a z^3+19 z^3 a^{-1} -5 a z- a^{-1} z^{-1} +19 z^8-42 z^6+22 z^4-3 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                   4   4 6                 5 1   -4 4               9 4     5 2             9 6       -3 0           10 8         2 -2         8 10           2 -4       6 9             -3 -6     3 8               5 -8   1 6                 -5 -10   3                   3 -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}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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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