L11a227

L11a226

L11a228

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Link Presentations

[edit Notes on L11a227's Link Presentations]

Planar diagram presentation	X8192 X20,9,21,10 X4758 X16,5,17,6 X14,18,15,17 X6,15,1,16 X22,14,7,13 X18,22,19,21 X2,11,3,12 X12,3,13,4 X10,19,11,20
Gauss code	{1, -9, 10, -3, 4, -6}, {3, -1, 2, -11, 9, -10, 7, -5, 6, -4, 5, -8, 11, -2, 8, -7}

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{u^2 v^4-5 u^2 v^3+8 u^2 v^2-3 u^2 v-2 u v^4+10 u v^3-15 u v^2+10 u v-2 u-3 v^3+8 v^2-5 v+1}{u v^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{23}{q^{9/2}}-\frac{24}{q^{7/2}}+\frac{21}{q^{5/2}}+q^{3/2}-\frac{16}{q^{3/2}}-\frac{1}{q^{19/2}}+\frac{3}{q^{17/2}}-\frac{8}{q^{15/2}}+\frac{14}{q^{13/2}}-\frac{20}{q^{11/2}}-5 \sqrt{q}+\frac{10}{\sqrt{q}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^9 z+a^9 z^{-1} -3 a^7 z^3-5 a^7 z-2 a^7 z^{-1} +3 a^5 z^5+7 a^5 z^3+5 a^5 z+2 a^5 z^{-1} -a^3 z^7-3 a^3 z^5-4 a^3 z^3-3 a^3 z-a^3 z^{-1} +a z^5+a z^3-a z }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^5 a^{11}+2 z^3 a^{11}-z a^{11}-3 z^6 a^{10}+4 z^4 a^{10}-z^2 a^{10}-6 z^7 a^9+9 z^5 a^9-8 z^3 a^9+5 z a^9-a^9 z^{-1} -8 z^8 a^8+12 z^6 a^8-11 z^4 a^8+4 z^2 a^8-6 z^9 a^7-z^7 a^7+19 z^5 a^7-26 z^3 a^7+12 z a^7-2 a^7 z^{-1} -2 z^{10} a^6-16 z^8 a^6+42 z^6 a^6-37 z^4 a^6+12 z^2 a^6-a^6-13 z^9 a^5+15 z^7 a^5+15 z^5 a^5-26 z^3 a^5+12 z a^5-2 a^5 z^{-1} -2 z^{10} a^4-17 z^8 a^4+48 z^6 a^4-34 z^4 a^4+8 z^2 a^4-7 z^9 a^3+5 z^7 a^3+16 z^5 a^3-14 z^3 a^3+5 z a^3-a^3 z^{-1} -9 z^8 a^2+20 z^6 a^2-11 z^4 a^2+z^2 a^2-5 z^7 a+10 z^5 a-4 z^3 a-z a-z^6+z^4 }[/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                     4   4 0                   6 1   -5 -2                 10 4     6 -4               12 7       -5 -6             12 9         3 -8           11 12           1 -10         9 12             -3 -12       5 11               6 -14     3 9                 -6 -16   1 6                   5 -18   2                     -2 -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}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/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=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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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