L10a52

L10a51

L10a53

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Link Presentations

[edit Notes on L10a52's Link Presentations]

Planar diagram presentation	X8192 X10,4,11,3 X20,10,7,9 X2738 X16,11,17,12 X12,5,13,6 X4,18,5,17 X14,20,15,19 X18,14,19,13 X6,15,1,16
Gauss code	{1, -4, 2, -7, 6, -10}, {4, -1, 3, -2, 5, -6, 9, -8, 10, -5, 7, -9, 8, -3}

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+6 t(1) t(2)^3-3 t(2)^3+4 t(1)^2 t(2)^2-7 t(1) t(2)^2+4 t(2)^2-3 t(1)^2 t(2)+6 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{ -4 q^{9/2}+\frac{1}{q^{9/2}}+8 q^{7/2}-\frac{4}{q^{7/2}}-12 q^{5/2}+\frac{7}{q^{5/2}}+15 q^{3/2}-\frac{12}{q^{3/2}}+q^{11/2}-16 \sqrt{q}+\frac{14}{\sqrt{q}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^7 a^{-1} +2 a z^5-4 z^5 a^{-1} +z^5 a^{-3} -a^3 z^3+5 a z^3-6 z^3 a^{-1} +2 z^3 a^{-3} -a^3 z+2 a z-3 z a^{-1} +z a^{-3} +a^3 z^{-1} -a z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -2 a z^9-2 z^9 a^{-1} -5 a^2 z^8-7 z^8 a^{-2} -12 z^8-4 a^3 z^7-9 a z^7-15 z^7 a^{-1} -10 z^7 a^{-3} -a^4 z^6+10 a^2 z^6+2 z^6 a^{-2} -8 z^6 a^{-4} +21 z^6+11 a^3 z^5+34 a z^5+40 z^5 a^{-1} +13 z^5 a^{-3} -4 z^5 a^{-5} +2 a^4 z^4-2 a^2 z^4+11 z^4 a^{-2} +8 z^4 a^{-4} -z^4 a^{-6} -2 z^4-9 a^3 z^3-27 a z^3-27 z^3 a^{-1} -7 z^3 a^{-3} +2 z^3 a^{-5} -a^4 z^2-2 a^2 z^2-6 z^2 a^{-2} -3 z^2 a^{-4} -4 z^2+a^3 z+5 a z+6 z a^{-1} +2 z a^{-3} -a^2+a^3 z^{-1} +a 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 12                     1 -1 10                   3   3 8                 5 1   -4 6               7 3     4 4             8 5       -3 2           8 7         1 0         7 9           2 -2       5 7             -2 -4     3 8               5 -6   1 4                 -3 -8   3                   3 -10 1                     -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=0 }[/math] [math]\displaystyle{ i=2 }[/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}^{4}\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^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\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^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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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