L11a269

L11a268

L11a270

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Link Presentations

[edit Notes on L11a269's Link Presentations]

Planar diagram presentation	X10,1,11,2 X2,11,3,12 X12,3,13,4 X4,9,5,10 X14,5,15,6 X22,20,9,19 X18,7,19,8 X6,17,7,18 X16,22,17,21 X20,16,21,15 X8,13,1,14
Gauss code	{1, -2, 3, -4, 5, -8, 7, -11}, {4, -1, 2, -3, 11, -5, 10, -9, 8, -7, 6, -10, 9, -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(1)^2 t(2)^5+t(1)^3 t(2)^4-4 t(1)^2 t(2)^4+3 t(1) t(2)^4-2 t(1)^3 t(2)^3+8 t(1)^2 t(2)^3-7 t(1) t(2)^3+2 t(2)^3+2 t(1)^3 t(2)^2-7 t(1)^2 t(2)^2+8 t(1) t(2)^2-2 t(2)^2+3 t(1)^2 t(2)-4 t(1) t(2)+t(2)+t(1)}{t(1)^{3/2} t(2)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{3/2}-3 \sqrt{q}+\frac{6}{\sqrt{q}}-\frac{11}{q^{3/2}}+\frac{15}{q^{5/2}}-\frac{18}{q^{7/2}}+\frac{18}{q^{9/2}}-\frac{16}{q^{11/2}}+\frac{12}{q^{13/2}}-\frac{8}{q^{15/2}}+\frac{3}{q^{17/2}}-\frac{1}{q^{19/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^7 z^5+3 a^7 z^3+4 a^7 z+2 a^7 z^{-1} -a^5 z^7-4 a^5 z^5-7 a^5 z^3-7 a^5 z-3 a^5 z^{-1} -a^3 z^7-4 a^3 z^5-6 a^3 z^3-3 a^3 z+a^3 z^{-1} +a z^5+3 a z^3+2 a z }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{11} z^5-2 a^{11} z^3+a^{11} z+3 a^{10} z^6-4 a^{10} z^4+a^{10} z^2+6 a^9 z^7-11 a^9 z^5+10 a^9 z^3-5 a^9 z+6 a^8 z^8-6 a^8 z^6-a^8 z^4+2 a^8 z^2+4 a^7 z^9-3 a^7 z^5-4 a^7 z^3+6 a^7 z-2 a^7 z^{-1} +a^6 z^{10}+10 a^6 z^8-21 a^6 z^6+17 a^6 z^4-9 a^6 z^2+3 a^6+7 a^5 z^9-10 a^5 z^7+9 a^5 z^5-15 a^5 z^3+12 a^5 z-3 a^5 z^{-1} +a^4 z^{10}+8 a^4 z^8-22 a^4 z^6+22 a^4 z^4-13 a^4 z^2+3 a^4+3 a^3 z^9-a^3 z^7-9 a^3 z^5+9 a^3 z^3-2 a^3 z-a^3 z^{-1} +4 a^2 z^8-9 a^2 z^6+5 a^2 z^4-a^2 z^2+a^2+3 a z^7-9 a z^5+8 a z^3-2 a z+z^6-3 z^4+2 z^2 }[/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 4                       1 -1 2                     2   2 0                   4 1   -3 -2                 7 2     5 -4               9 5       -4 -6             9 6         3 -8           9 9           0 -10         7 9             -2 -12       5 9               4 -14     3 7                 -4 -16     5                   5 -18 1 3                     -2 -20 1                       1

Integral Khovanov Homology (db, data source)

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