L11a545

L11a544

L11a546

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Link Presentations

[edit Notes on L11a545's Link Presentations]

Planar diagram presentation	X6172 X14,5,15,6 X12,4,13,3 X2,9,3,10 X18,7,19,8 X8,17,9,18 X10,13,5,14 X22,20,17,19 X16,21,11,22 X20,11,21,12 X4,16,1,15
Gauss code	{1, -4, 3, -11}, {2, -1, 5, -6, 4, -7}, {10, -3, 7, -2, 11, -9}, {6, -5, 8, -10, 9, -8}

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 v^2 w^2 x-2 u v^2 w^2-2 u v^2 w x+3 u v^2 w+u v^2 x-u v^2-2 u v w^2 x+2 u v w^2+5 u v w x-5 u v w-3 u v x+2 u v-2 u w x+2 u w+2 u x-u-v^2 w^2 x+2 v^2 w^2+2 v^2 w x-2 v^2 w+2 v w^2 x-3 v w^2-5 v w x+5 v w+2 v x-2 v-w^2 x+w^2+3 w x-2 w-2 x+1}{\sqrt{u} v w \sqrt{x}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{21}{q^{9/2}}-\frac{25}{q^{7/2}}+\frac{20}{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{13}{q^{13/2}}-\frac{21}{q^{11/2}}-5 \sqrt{q}+\frac{10}{\sqrt{q}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z a^9+a^9 z^{-1} +a^9 z^{-3} -3 z^3 a^7-6 z a^7-6 a^7 z^{-1} -3 a^7 z^{-3} +3 z^5 a^5+8 z^3 a^5+10 z a^5+9 a^5 z^{-1} +3 a^5 z^{-3} -z^7 a^3-3 z^5 a^3-4 z^3 a^3-4 z a^3-4 a^3 z^{-1} -a^3 z^{-3} +z^5 a+z^3 a-z a }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{11} z^5-2 a^{11} z^3+a^{11} z+3 a^{10} z^6-4 a^{10} z^4+a^{10} z^2+6 a^9 z^7-10 a^9 z^5+12 a^9 z^3-a^9 z^{-3} -11 a^9 z+5 a^9 z^{-1} +7 a^8 z^8-6 a^8 z^6-4 a^8 z^4+14 a^8 z^2+3 a^8 z^{-2} -10 a^8+5 a^7 z^9+7 a^7 z^7-35 a^7 z^5+50 a^7 z^3-3 a^7 z^{-3} -33 a^7 z+12 a^7 z^{-1} +2 a^6 z^{10}+13 a^6 z^8-25 a^6 z^6+26 a^6 z^2+6 a^6 z^{-2} -19 a^6+12 a^5 z^9-10 a^5 z^7-27 a^5 z^5+46 a^5 z^3-3 a^5 z^{-3} -33 a^5 z+12 a^5 z^{-1} +2 a^4 z^{10}+15 a^4 z^8-37 a^4 z^6+11 a^4 z^4+14 a^4 z^2+3 a^4 z^{-2} -10 a^4+7 a^3 z^9-6 a^3 z^7-13 a^3 z^5+14 a^3 z^3-a^3 z^{-3} -11 a^3 z+5 a^3 z^{-1} +9 a^2 z^8-20 a^2 z^6+10 a^2 z^4+a^2 z^2+5 a z^7-10 a z^5+4 a z^3+a z+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 8       -4 -6             13 8         5 -8           10 14           4 -10         11 11             0 -12       4 12               8 -14     4 9                 -5 -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}^{4} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} }[/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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