L11a462

L11a461

L11a463

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Link Presentations

[edit Notes on L11a462's Link Presentations]

Planar diagram presentation	X6172 X14,4,15,3 X16,5,17,6 X18,14,19,13 X12,15,5,16 X10,19,11,20 X8,21,9,22 X20,7,21,8 X22,9,13,10 X2,11,3,12 X4,18,1,17
Gauss code	{1, -10, 2, -11}, {3, -1, 8, -7, 9, -6, 10, -5}, {4, -2, 5, -3, 11, -4, 6, -8, 7, -9}

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(3)^3 t(2)^3-t(3)^3 t(2)^3-2 t(1) t(3)^2 t(2)^3+2 t(3)^2 t(2)^3+t(1) t(3) t(2)^3-2 t(1) t(3)^3 t(2)^2+2 t(3)^3 t(2)^2+6 t(1) t(3)^2 t(2)^2-6 t(3)^2 t(2)^2+t(1) t(2)^2-5 t(1) t(3) t(2)^2+4 t(3) t(2)^2-t(3)^3 t(2)-4 t(1) t(3)^2 t(2)+5 t(3)^2 t(2)-2 t(1) t(2)+6 t(1) t(3) t(2)-6 t(3) t(2)+2 t(2)-t(3)^2+t(1)-2 t(1) t(3)+2 t(3)-1}{\sqrt{t(1)} t(2)^{3/2} t(3)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q+4-7 q^{-1} +13 q^{-2} -17 q^{-3} +21 q^{-4} -21 q^{-5} +19 q^{-6} -14 q^{-7} +10 q^{-8} -4 q^{-9} + q^{-10} }[/math] (db)
Signature	-4 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^8 z^4+2 a^8 z^2+a^8 z^{-2} +2 a^8-2 a^6 z^6-7 a^6 z^4-8 a^6 z^2-2 a^6 z^{-2} -5 a^6+a^4 z^8+5 a^4 z^6+9 a^4 z^4+6 a^4 z^2+a^4 z^{-2} +a^4-a^2 z^6-3 a^2 z^4-a^2 z^2+2 a^2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{12} z^4+4 a^{11} z^5+10 a^{10} z^6-10 a^{10} z^4+6 a^{10} z^2-2 a^{10}+14 a^9 z^7-18 a^9 z^5+6 a^9 z^3+a^9 z+13 a^8 z^8-17 a^8 z^6+6 a^8 z^4-6 a^8 z^2-a^8 z^{-2} +3 a^8+7 a^7 z^9+4 a^7 z^7-32 a^7 z^5+22 a^7 z^3-8 a^7 z+2 a^7 z^{-1} +2 a^6 z^{10}+16 a^6 z^8-50 a^6 z^6+42 a^6 z^4-22 a^6 z^2-2 a^6 z^{-2} +9 a^6+12 a^5 z^9-26 a^5 z^7+4 a^5 z^5+14 a^5 z^3-8 a^5 z+2 a^5 z^{-1} +2 a^4 z^{10}+7 a^4 z^8-38 a^4 z^6+42 a^4 z^4-14 a^4 z^2-a^4 z^{-2} +3 a^4+5 a^3 z^9-15 a^3 z^7+11 a^3 z^5+a^3 z+4 a^2 z^8-15 a^2 z^6+17 a^2 z^4-4 a^2 z^2-2 a^2+a z^7-3 a z^5+2 a z^3 }[/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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 3                       1 -1 1                     3   3 -1                   4 1   -3 -3                 9 3     6 -5               10 6       -4 -7             11 7         4 -9           10 10           0 -11         9 11             -2 -13       6 11               5 -15     4 8                 -4 -17     6                   6 -19 1 4                     -3 -21 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=-8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=3 }[/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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