L11a350

L11a349

L11a351

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Link Presentations

[edit Notes on L11a350's Link Presentations]

Planar diagram presentation	X12,1,13,2 X8493 X18,14,19,13 X22,20,11,19 X20,7,21,8 X6,21,7,22 X4,15,5,16 X14,5,15,6 X16,10,17,9 X2,11,3,12 X10,18,1,17
Gauss code	{1, -10, 2, -7, 8, -6, 5, -2, 9, -11}, {10, -1, 3, -8, 7, -9, 11, -3, 4, -5, 6, -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{t(2)^3 t(1)^3-3 t(2)^2 t(1)^3+4 t(2) t(1)^3-2 t(1)^3-3 t(2)^3 t(1)^2+8 t(2)^2 t(1)^2-8 t(2) t(1)^2+4 t(1)^2+4 t(2)^3 t(1)-8 t(2)^2 t(1)+8 t(2) t(1)-3 t(1)-2 t(2)^3+4 t(2)^2-3 t(2)+1}{t(1)^{3/2} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{11/2}-4 q^{9/2}+9 q^{7/2}-15 q^{5/2}+19 q^{3/2}-21 \sqrt{q}+\frac{21}{\sqrt{q}}-\frac{18}{q^{3/2}}+\frac{12}{q^{5/2}}-\frac{8}{q^{7/2}}+\frac{3}{q^{9/2}}-\frac{1}{q^{11/2}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^7 a^{-1} +3 a z^5-4 z^5 a^{-1} +z^5 a^{-3} -3 a^3 z^3+8 a z^3-8 z^3 a^{-1} +2 z^3 a^{-3} +a^5 z-5 a^3 z+6 a z-6 z a^{-1} +2 z a^{-3} +a^5 z^{-1} -a^3 z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -2 a^2 z^{10}-2 z^{10}-4 a^3 z^9-12 a z^9-8 z^9 a^{-1} -3 a^4 z^8-6 a^2 z^8-14 z^8 a^{-2} -17 z^8-a^5 z^7+9 a^3 z^7+25 a z^7+z^7 a^{-1} -14 z^7 a^{-3} +10 a^4 z^6+32 a^2 z^6+22 z^6 a^{-2} -9 z^6 a^{-4} +53 z^6+4 a^5 z^5+a^3 z^5-a z^5+26 z^5 a^{-1} +20 z^5 a^{-3} -4 z^5 a^{-5} -10 a^4 z^4-32 a^2 z^4-10 z^4 a^{-2} +7 z^4 a^{-4} -z^4 a^{-6} -40 z^4-6 a^5 z^3-11 a^3 z^3-14 a z^3-21 z^3 a^{-1} -11 z^3 a^{-3} +z^3 a^{-5} +2 a^4 z^2+9 a^2 z^2+2 z^2 a^{-2} -2 z^2 a^{-4} +11 z^2+4 a^5 z+5 a^3 z+4 a z+5 z a^{-1} +2 z a^{-3} +a^4-a^5 z^{-1} -a^3 z^{-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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 12                       1 -1 10                     3   3 8                   6 1   -5 6                 9 3     6 4               10 6       -4 2             11 9         2 0           11 11           0 -2         7 10             -3 -4       5 11               6 -6     3 7                 -4 -8   1 6                   5 -10   2                     -2 -12 1                       1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=0 }[/math] [math]\displaystyle{ i=2 }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=5 }[/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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