L10a61

L10a60

L10a62

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Link Presentations

[edit Notes on L10a61's Link Presentations]

Planar diagram presentation	X8192 X10,4,11,3 X20,10,7,9 X2738 X18,14,19,13 X6,12,1,11 X16,20,17,19 X4,16,5,15 X14,6,15,5 X12,18,13,17
Gauss code	{1, -4, 2, -8, 9, -6}, {4, -1, 3, -2, 6, -10, 5, -9, 8, -7, 10, -5, 7, -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) t(2)^4-t(2)^4+2 t(1)^2 t(2)^3-6 t(1) t(2)^3+3 t(2)^3-4 t(1)^2 t(2)^2+9 t(1) t(2)^2-4 t(2)^2+3 t(1)^2 t(2)-6 t(1) t(2)+2 t(2)-t(1)^2+t(1)}{t(1) t(2)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^{19/2}+4 q^{17/2}-7 q^{15/2}+11 q^{13/2}-14 q^{11/2}+14 q^{9/2}-14 q^{7/2}+10 q^{5/2}-7 q^{3/2}+3 \sqrt{q}-\frac{1}{\sqrt{q}} }[/math] (db)
Signature	3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^5 a^{-3} -2 z^5 a^{-5} +z^3 a^{-1} -5 z^3 a^{-5} +3 z^3 a^{-7} +z a^{-1} +3 z a^{-3} -6 z a^{-5} +4 z a^{-7} -z a^{-9} +2 a^{-3} z^{-1} -3 a^{-5} z^{-1} + a^{-7} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^5 a^{-11} -z^3 a^{-11} +4 z^6 a^{-10} -7 z^4 a^{-10} +3 z^2 a^{-10} +6 z^7 a^{-9} -10 z^5 a^{-9} +5 z^3 a^{-9} -2 z a^{-9} +4 z^8 a^{-8} +3 z^6 a^{-8} -18 z^4 a^{-8} +12 z^2 a^{-8} - a^{-8} +z^9 a^{-7} +14 z^7 a^{-7} -34 z^5 a^{-7} +28 z^3 a^{-7} -10 z a^{-7} + a^{-7} z^{-1} +8 z^8 a^{-6} -5 z^6 a^{-6} -13 z^4 a^{-6} +14 z^2 a^{-6} -3 a^{-6} +z^9 a^{-5} +13 z^7 a^{-5} -33 z^5 a^{-5} +32 z^3 a^{-5} -15 z a^{-5} +3 a^{-5} z^{-1} +4 z^8 a^{-4} -z^6 a^{-4} -7 z^4 a^{-4} +7 z^2 a^{-4} -3 a^{-4} +5 z^7 a^{-3} -9 z^5 a^{-3} +8 z^3 a^{-3} -6 z a^{-3} +2 a^{-3} z^{-1} +3 z^6 a^{-2} -5 z^4 a^{-2} +2 z^2 a^{-2} +z^5 a^{-1} -2 z^3 a^{-1} +z a^{-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   \ -2 -1 0 1 2 3 4 5 6 7 8 χ 20                     1 1 18                   3   -3 16                 4 1   3 14               7 3     -4 12             7 4       3 10           7 7         0 8         7 7           0 6       4 8             4 4     3 6               -3 2   1 5                 4 0   2                   -2 -2 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=4 }[/math] [math]\displaystyle{ r=-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}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=7 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=8 }[/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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