L11a297

L11a296

L11a298

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Link Presentations

[edit Notes on L11a297's Link Presentations]

Planar diagram presentation	X10,1,11,2 X12,3,13,4 X14,21,15,22 X16,5,17,6 X18,8,19,7 X20,17,21,18 X4,15,5,16 X6,20,7,19 X22,13,9,14 X2,9,3,10 X8,11,1,12
Gauss code	{1, -10, 2, -7, 4, -8, 5, -11}, {10, -1, 11, -2, 9, -3, 7, -4, 6, -5, 8, -6, 3, -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{2 u^3 v^2-3 u^3 v+u^3+2 u^2 v^3-7 u^2 v^2+7 u^2 v-3 u^2-3 u v^3+7 u v^2-7 u v+2 u+v^3-3 v^2+2 v}{u^{3/2} v^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{16}{q^{9/2}}-\frac{14}{q^{7/2}}+\frac{10}{q^{5/2}}-\frac{7}{q^{3/2}}+\frac{1}{q^{21/2}}-\frac{3}{q^{19/2}}+\frac{6}{q^{17/2}}-\frac{10}{q^{15/2}}+\frac{13}{q^{13/2}}-\frac{16}{q^{11/2}}-\sqrt{q}+\frac{3}{\sqrt{q}} }[/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+2 z^5 a^5+4 z^3 a^5+3 z a^5+a^5 z^{-1} +z^5 a^3-3 z a^3-a^3 z^{-1} -z^3 a-z a }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{12} z^6-3 a^{12} z^4+2 a^{12} z^2+3 a^{11} z^7-9 a^{11} z^5+7 a^{11} z^3-2 a^{11} z+4 a^{10} z^8-10 a^{10} z^6+6 a^{10} z^4-2 a^{10} z^2+3 a^9 z^9-3 a^9 z^7-6 a^9 z^5+7 a^9 z^3-3 a^9 z+a^8 z^{10}+6 a^8 z^8-19 a^8 z^6+17 a^8 z^4-5 a^8 z^2+6 a^7 z^9-9 a^7 z^7-a^7 z^5+10 a^7 z^3-3 a^7 z+a^6 z^{10}+7 a^6 z^8-18 a^6 z^6+16 a^6 z^4-2 a^6 z^2+3 a^5 z^9+2 a^5 z^7-14 a^5 z^5+18 a^5 z^3-8 a^5 z+a^5 z^{-1} +5 a^4 z^8-7 a^4 z^6+3 a^4 z^4-a^4+5 a^3 z^7-9 a^3 z^5+6 a^3 z^3-5 a^3 z+a^3 z^{-1} +3 a^2 z^6-5 a^2 z^4+a^2 z^2+a z^5-2 a z^3+a z }[/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 6         -2 -10           8 8           0 -12         6 9             3 -14       4 7               -3 -16     2 6                 4 -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}^{6}\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^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} }[/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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