L11a217

L11a216

L11a218

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Link Presentations

[edit Notes on L11a217's Link Presentations]

Planar diagram presentation	X8192 X12,3,13,4 X16,6,17,5 X22,13,7,14 X18,15,19,16 X14,21,15,22 X20,10,21,9 X4,18,5,17 X10,20,11,19 X2738 X6,11,1,12
Gauss code	{1, -10, 2, -8, 3, -11}, {10, -1, 7, -9, 11, -2, 4, -6, 5, -3, 8, -5, 9, -7, 6, -4}

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^2 v^3-7 u^2 v^2+6 u^2 v-u^2+u v^4-9 u v^3+17 u v^2-9 u v+u-v^4+6 v^3-7 v^2+2 v}{u v^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{15}{q^{9/2}}-q^{7/2}+\frac{19}{q^{7/2}}+3 q^{5/2}-\frac{23}{q^{5/2}}-8 q^{3/2}+\frac{22}{q^{3/2}}+\frac{1}{q^{15/2}}-\frac{4}{q^{13/2}}+\frac{9}{q^{11/2}}+14 \sqrt{q}-\frac{19}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^7 (-z)+3 a^5 z^3+2 a^5 z+a^5 z^{-1} -2 a^3 z^5-3 a^3 z^3-5 a^3 z-2 a^3 z^{-1} -z a^{-3} -a z^5+3 a z^3+2 z^3 a^{-1} +5 a z+2 a z^{-1} -z a^{-1} - a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -2 a^4 z^{10}-2 a^2 z^{10}-6 a^5 z^9-12 a^3 z^9-6 a z^9-7 a^6 z^8-13 a^4 z^8-14 a^2 z^8-8 z^8-4 a^7 z^7+5 a^5 z^7+16 a^3 z^7+a z^7-6 z^7 a^{-1} -a^8 z^6+15 a^6 z^6+39 a^4 z^6+39 a^2 z^6-3 z^6 a^{-2} +13 z^6+9 a^7 z^5+13 a^5 z^5+9 a^3 z^5+15 a z^5+9 z^5 a^{-1} -z^5 a^{-3} +2 a^8 z^4-8 a^6 z^4-30 a^4 z^4-37 a^2 z^4+4 z^4 a^{-2} -13 z^4-6 a^7 z^3-14 a^5 z^3-22 a^3 z^3-23 a z^3-7 z^3 a^{-1} +2 z^3 a^{-3} -a^8 z^2+2 a^6 z^2+8 a^4 z^2+13 a^2 z^2-z^2 a^{-2} +7 z^2+2 a^7 z+5 a^5 z+10 a^3 z+12 a z+4 z a^{-1} -z a^{-3} -a^2-a^5 z^{-1} -2 a^3 z^{-1} -2 a z^{-1} - a^{-1} z^{-1} }[/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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 8                       1 1 6                     2   -2 4                   6 1   5 2                 8 2     -6 0               11 6       5 -2             12 9         -3 -4           11 10           1 -6         9 13             4 -8       6 10               -4 -10     3 9                 6 -12   1 6                   -5 -14   3                     3 -16 1                       -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-2 }[/math] [math]\displaystyle{ i=0 }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=4 }[/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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