L10a130

L10a129

L10a131

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Link Presentations

[edit Notes on L10a130's Link Presentations]

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

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{2 t(3)^2 t(2)^2-t(1) t(2)^2+2 t(1) t(3) t(2)^2-3 t(3) t(2)^2+t(2)^2+2 t(1) t(3)^2 t(2)-3 t(3)^2 t(2)+3 t(1) t(2)-5 t(1) t(3) t(2)+5 t(3) t(2)-2 t(2)-t(1) t(3)^2+t(3)^2-2 t(1)+3 t(1) t(3)-2 t(3)}{\sqrt{t(1)} t(2) t(3)} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{-2} -3 q^{-3} +7 q^{-4} -9 q^{-5} +13 q^{-6} -12 q^{-7} +12 q^{-8} -9 q^{-9} +6 q^{-10} -3 q^{-11} + q^{-12} }[/math] (db)
Signature	-4 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^{12}-3 a^{10} z^2+a^{10} z^{-2} -2 a^{10}+2 a^8 z^4+a^8 z^2-2 a^8 z^{-2} -3 a^8+3 a^6 z^4+6 a^6 z^2+a^6 z^{-2} +4 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}-9 z^5 a^{13}+8 z^3 a^{13}-2 z a^{13}+3 z^8 a^{12}-4 z^6 a^{12}-6 z^4 a^{12}+8 z^2 a^{12}-a^{12}+z^9 a^{11}+8 z^7 a^{11}-27 z^5 a^{11}+22 z^3 a^{11}-6 z a^{11}+7 z^8 a^{10}-9 z^6 a^{10}-6 z^4 a^{10}+7 z^2 a^{10}+a^{10} z^{-2} -3 a^{10}+z^9 a^9+11 z^7 a^9-26 z^5 a^9+15 z^3 a^9-2 a^9 z^{-1} +4 z^8 a^8+2 z^6 a^8-13 z^4 a^8+12 z^2 a^8+2 a^8 z^{-2} -6 a^8+6 z^7 a^7-5 z^5 a^7-z^3 a^7+4 z a^7-2 a^7 z^{-1} +6 z^6 a^6-9 z^4 a^6+9 z^2 a^6+a^6 z^{-2} -4 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               5 3     -2 -11             8 4       4 -13           5 6         1 -15         7 7           0 -17       4 7             3 -19     2 5               -3 -21   1 4                 3 -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}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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}^{7}\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^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/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=-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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