L11a191

L11a190

L11a192

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Link Presentations

[edit Notes on L11a191's Link Presentations]

Planar diagram presentation	X8192 X10,3,11,4 X16,6,17,5 X20,11,21,12 X22,13,7,14 X12,21,13,22 X18,15,19,16 X14,19,15,20 X4,18,5,17 X2738 X6,9,1,10
Gauss code	{1, -10, 2, -9, 3, -11}, {10, -1, 11, -2, 4, -6, 5, -8, 7, -3, 9, -7, 8, -4, 6, -5}

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(1) t(2)^4-2 t(2)^4+2 t(1)^2 t(2)^3-6 t(1) t(2)^3+3 t(2)^3-3 t(1)^2 t(2)^2+7 t(1) t(2)^2-3 t(2)^2+3 t(1)^2 t(2)-6 t(1) t(2)+2 t(2)-2 t(1)^2+2 t(1)}{t(1) t(2)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\sqrt{q}+\frac{2}{\sqrt{q}}-\frac{5}{q^{3/2}}+\frac{8}{q^{5/2}}-\frac{12}{q^{7/2}}+\frac{13}{q^{9/2}}-\frac{14}{q^{11/2}}+\frac{12}{q^{13/2}}-\frac{9}{q^{15/2}}+\frac{6}{q^{17/2}}-\frac{3}{q^{19/2}}+\frac{1}{q^{21/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^3 a^9-z a^9+z^5 a^7+z^3 a^7-z a^7-a^7 z^{-1} +2 z^5 a^5+5 z^3 a^5+5 z a^5+3 a^5 z^{-1} +z^5 a^3+z^3 a^3-2 z a^3-2 a^3 z^{-1} -z^3 a-2 z a }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^6 a^{12}+3 z^4 a^{12}-2 z^2 a^{12}-3 z^7 a^{11}+9 z^5 a^{11}-6 z^3 a^{11}-4 z^8 a^{10}+11 z^6 a^{10}-8 z^4 a^{10}+3 z^2 a^{10}-3 z^9 a^9+6 z^7 a^9-5 z^5 a^9+8 z^3 a^9-3 z a^9-z^{10} a^8-3 z^8 a^8+9 z^6 a^8-4 z^4 a^8-z^2 a^8+a^8-5 z^9 a^7+11 z^7 a^7-11 z^5 a^7+3 z^3 a^7+2 z a^7-a^7 z^{-1} -z^{10} a^6-2 z^8 a^6+3 z^6 a^6+2 z^4 a^6-9 z^2 a^6+3 a^6-2 z^9 a^5-z^7 a^5+10 z^5 a^5-20 z^3 a^5+13 z a^5-3 a^5 z^{-1} -3 z^8 a^4+4 z^6 a^4-z^4 a^4-4 z^2 a^4+3 a^4-3 z^7 a^3+6 z^5 a^3-6 z^3 a^3+6 z a^3-2 a^3 z^{-1} -2 z^6 a^2+4 z^4 a^2-z^2 a^2-z^5 a+3 z^3 a-2 z a }[/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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 2                       1 1 0                     1   -1 -2                   4 1   3 -4                 5 2     -3 -6               7 3       4 -8             7 6         -1 -10           7 6           1 -12         5 7             2 -14       4 7               -3 -16     2 5                 3 -18   1 4                   -3 -20   2                     2 -22 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=-9 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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}^{7}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=2 }[/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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