L10a164

L10a163

L10a165

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Link Presentations

[edit Notes on L10a164's Link Presentations]

Planar diagram presentation	X8192 X14,3,15,4 X12,15,7,16 X10,19,11,20 X16,9,17,10 X20,11,13,12 X18,5,19,6 X2738 X4,13,5,14 X6,17,1,18
Gauss code	{1, -8, 2, -9, 7, -10}, {8, -1, 5, -4, 6, -3}, {9, -2, 3, -5, 10, -7, 4, -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(2)^2 t(3)^3+t(1) t(2) t(3)^3-t(2) t(3)^3-t(1)^2 t(3)^2+2 t(1) t(2)^2 t(3)^2-2 t(2)^2 t(3)^2+2 t(1) t(3)^2+2 t(1)^2 t(2) t(3)^2-4 t(1) t(2) t(3)^2+3 t(2) t(3)^2-t(3)^2+2 t(1)^2 t(3)+t(1)^2 t(2)^2 t(3)-2 t(1) t(2)^2 t(3)+t(2)^2 t(3)-2 t(1) t(3)-3 t(1)^2 t(2) t(3)+4 t(1) t(2) t(3)-2 t(2) t(3)-t(1)^2+t(1)^2 t(2)-t(1) t(2)}{t(1) t(2) t(3)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{-2} -3 q^{-3} +7 q^{-4} -10 q^{-5} +13 q^{-6} -13 q^{-7} +13 q^{-8} -9 q^{-9} +7 q^{-10} -3 q^{-11} + q^{-12} }[/math] (db)
Signature	-4 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^{12} z^{-2} +a^{12}-4 a^{10} z^2-2 a^{10} z^{-2} -6 a^{10}+3 a^8 z^4+6 a^8 z^2+a^8 z^{-2} +4 a^8+3 a^6 z^4+5 a^6 z^2+a^6+a^4 z^4+a^4 z^2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^6 a^{14}-3 z^4 a^{14}+3 z^2 a^{14}-a^{14}+3 z^7 a^{13}-8 z^5 a^{13}+5 z^3 a^{13}+4 z^8 a^{12}-9 z^6 a^{12}+4 z^4 a^{12}-3 z^2 a^{12}-a^{12} z^{-2} +5 a^{12}+2 z^9 a^{11}+3 z^7 a^{11}-18 z^5 a^{11}+17 z^3 a^{11}-9 z a^{11}+2 a^{11} z^{-1} +10 z^8 a^{10}-25 z^6 a^{10}+26 z^4 a^{10}-23 z^2 a^{10}-2 a^{10} z^{-2} +11 a^{10}+2 z^9 a^9+7 z^7 a^9-22 z^5 a^9+19 z^3 a^9-9 z a^9+2 a^9 z^{-1} +6 z^8 a^8-9 z^6 a^8+10 z^4 a^8-10 z^2 a^8-a^8 z^{-2} +5 a^8+7 z^7 a^7-9 z^5 a^7+5 z^3 a^7+6 z^6 a^6-8 z^4 a^6+6 z^2 a^6-a^6+3 z^5 a^5-2 z^3 a^5+z^4 a^4-z^2 a^4 }[/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   \ -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -3                     1 1 -5                   3 1 -2 -7                 4     4 -9               6 3     -3 -11             7 4       3 -13           6 6         0 -15         7 7           0 -17       4 8             4 -19     3 5               -2 -21   1 5                 4 -23   2                   -2 -25 1                     1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-5 }[/math] [math]\displaystyle{ i=-3 }[/math] [math]\displaystyle{ r=-10 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-9 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/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=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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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