L11a189

L11a188

L11a190

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Link Presentations

[edit Notes on L11a189's Link Presentations]

Planar diagram presentation	X8192 X10,3,11,4 X16,6,17,5 X18,11,19,12 X22,15,7,16 X12,21,13,22 X20,13,21,14 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, 7, -8, 5, -3, 9, -4, 8, -7, 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)^3-7 t(1) t(2)^3+3 t(2)^3-3 t(1)^2 t(2)^2+9 t(1) t(2)^2-3 t(2)^2+3 t(1)^2 t(2)-7 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{ \frac{15}{q^{9/2}}-\frac{14}{q^{7/2}}+\frac{10}{q^{5/2}}-\frac{7}{q^{3/2}}+\frac{1}{q^{21/2}}-\frac{2}{q^{19/2}}+\frac{5}{q^{17/2}}-\frac{9}{q^{15/2}}+\frac{12}{q^{13/2}}-\frac{15}{q^{11/2}}-\sqrt{q}+\frac{3}{\sqrt{q}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^3 a^9-2 z a^9-a^9 z^{-1} +z^5 a^7+2 z^3 a^7+3 z a^7+2 a^7 z^{-1} +2 z^5 a^5+4 z^3 a^5+2 z a^5+z^5 a^3-3 z a^3-a^3 z^{-1} -z^3 a-z a }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^6 a^{12}+4 z^4 a^{12}-4 z^2 a^{12}-2 z^7 a^{11}+6 z^5 a^{11}-4 z^3 a^{11}-3 z^8 a^{10}+9 z^6 a^{10}-11 z^4 a^{10}+9 z^2 a^{10}-2 a^{10}-3 z^9 a^9+9 z^7 a^9-17 z^5 a^9+20 z^3 a^9-7 z a^9+a^9 z^{-1} -z^{10} a^8-4 z^8 a^8+17 z^6 a^8-28 z^4 a^8+22 z^2 a^8-5 a^8-6 z^9 a^7+15 z^7 a^7-22 z^5 a^7+19 z^3 a^7-10 z a^7+2 a^7 z^{-1} -z^{10} a^6-6 z^8 a^6+17 z^6 a^6-20 z^4 a^6+9 z^2 a^6-3 a^6-3 z^9 a^5-z^7 a^5+11 z^5 a^5-13 z^3 a^5+3 z a^5-5 z^8 a^4+7 z^6 a^4-2 z^4 a^4-z^2 a^4+a^4-5 z^7 a^3+9 z^5 a^3-6 z^3 a^3+5 z a^3-a^3 z^{-1} -3 z^6 a^2+5 z^4 a^2-z^2 a^2-z^5 a+2 z^3 a-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                     2   -2 -2                   5 1   4 -4                 6 3     -3 -6               8 4       4 -8             8 7         -1 -10           7 7           0 -12         5 8             3 -14       4 7               -3 -16     1 5                 4 -18   1 4                   -3 -20   1                     1 -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}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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}^{8}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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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