L11a352

L11a351

L11a353

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Link Presentations

[edit Notes on L11a352's Link Presentations]

Planar diagram presentation	X12,1,13,2 X8493 X14,5,15,6 X20,7,21,8 X16,13,17,14 X18,10,19,9 X10,11,1,12 X6,15,7,16 X4,21,5,22 X22,17,11,18 X2,20,3,19
Gauss code	{1, -11, 2, -9, 3, -8, 4, -2, 6, -7}, {7, -1, 5, -3, 8, -5, 10, -6, 11, -4, 9, -10}

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^3-4 u^3 v^2+3 u^3 v-u^3-5 u^2 v^3+13 u^2 v^2-11 u^2 v+4 u^2+4 u v^3-11 u v^2+13 u v-5 u-v^3+3 v^2-4 v+2}{u^{3/2} v^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{28}{q^{9/2}}-\frac{28}{q^{7/2}}+\frac{23}{q^{5/2}}+q^{3/2}-\frac{17}{q^{3/2}}-\frac{1}{q^{19/2}}+\frac{5}{q^{17/2}}-\frac{12}{q^{15/2}}+\frac{19}{q^{13/2}}-\frac{25}{q^{11/2}}-4 \sqrt{q}+\frac{9}{\sqrt{q}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^5 a^7+z^3 a^7-z^7 a^5-2 z^5 a^5+2 z a^5+a^5 z^{-1} -z^7 a^3-3 z^5 a^3-5 z^3 a^3-5 z a^3-a^3 z^{-1} +z^5 a+2 z^3 a+z a }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{11} z^5+5 a^{10} z^6-3 a^{10} z^4+12 a^9 z^7-15 a^9 z^5+5 a^9 z^3-a^9 z+16 a^8 z^8-24 a^8 z^6+11 a^8 z^4-3 a^8 z^2+11 a^7 z^9-3 a^7 z^7-20 a^7 z^5+13 a^7 z^3-2 a^7 z+3 a^6 z^{10}+24 a^6 z^8-58 a^6 z^6+36 a^6 z^4-6 a^6 z^2+18 a^5 z^9-24 a^5 z^7-8 a^5 z^5+19 a^5 z^3-7 a^5 z+a^5 z^{-1} +3 a^4 z^{10}+15 a^4 z^8-43 a^4 z^6+31 a^4 z^4-5 a^4 z^2-a^4+7 a^3 z^9-5 a^3 z^7-13 a^3 z^5+18 a^3 z^3-8 a^3 z+a^3 z^{-1} +7 a^2 z^8-13 a^2 z^6+7 a^2 z^4-a^2 z^2+4 a z^7-9 a z^5+7 a z^3-2 a z+z^6-2 z^4+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                     3   3 0                   6 1   -5 -2                 11 3     8 -4               13 7       -6 -6             15 10         5 -8           13 13           0 -10         12 15             -3 -12       8 14               6 -14     4 11                 -7 -16   1 8                   7 -18   4                     -4 -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{ 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}^{8}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{15}\oplus{\mathbb Z}_2^{13} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{15} }[/math] [math]\displaystyle{ {\mathbb Z}^{15} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{13} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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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