L10a170

L10a169

L10a171

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Link Presentations

[edit Notes on L10a170's Link Presentations]

Planar diagram presentation	X6172 X12,6,13,5 X8493 X2,16,3,15 X16,7,17,8 X14,19,11,20 X20,13,15,14 X18,10,19,9 X10,12,5,11 X4,17,1,18
Gauss code	{1, -4, 3, -10}, {9, -2, 7, -6}, {2, -1, 5, -3, 8, -9}, {4, -5, 10, -8, 6, -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-1) (w-1) (x-1) (v w x+v (-w)+v-w x+x-1)}{\sqrt{u} \sqrt{v} w x} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -4 q^{9/2}+\frac{1}{q^{9/2}}+8 q^{7/2}-\frac{4}{q^{7/2}}-13 q^{5/2}+\frac{7}{q^{5/2}}+14 q^{3/2}-\frac{13}{q^{3/2}}+q^{11/2}-18 \sqrt{q}+\frac{13}{\sqrt{q}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^5 a^{-3} -a^3 z^3+2 z^3 a^{-3} +a^3 z^{-3} - a^{-3} z^{-3} -a^3 z+z a^{-3} +a^3 z^{-1} - a^{-3} z^{-1} -z^7 a^{-1} +2 a z^5-4 z^5 a^{-1} +5 a z^3-6 z^3 a^{-1} -3 a z^{-3} +3 a^{-1} z^{-3} +2 a z-2 z a^{-1} -3 a z^{-1} +3 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^4 a^{-6} +4 z^5 a^{-5} -2 z^3 a^{-5} +a^4 z^6+8 z^6 a^{-4} -2 a^4 z^4-7 z^4 a^{-4} +a^4 z^2+z^2 a^{-4} +4 a^3 z^7+11 z^7 a^{-3} -11 a^3 z^5-17 z^5 a^{-3} +10 a^3 z^3+12 z^3 a^{-3} +a^3 z^{-3} + a^{-3} z^{-3} -3 a^3 z-3 z a^{-3} -2 a^3 z^{-1} -2 a^{-3} z^{-1} +5 a^2 z^8+8 z^8 a^{-2} -9 a^2 z^6-6 z^6 a^{-2} -a^2 z^4-6 z^4 a^{-2} +3 a^2 z^2+3 z^2 a^{-2} -3 a^2 z^{-2} -3 a^{-2} z^{-2} +4 a^2+4 a^{-2} +2 a z^9+2 z^9 a^{-1} +11 a z^7+18 z^7 a^{-1} -43 a z^5-53 z^5 a^{-1} +40 a z^3+44 z^3 a^{-1} +3 a z^{-3} +3 a^{-1} z^{-3} -11 a z-11 z a^{-1} -3 a z^{-1} -3 a^{-1} z^{-1} +13 z^8-24 z^6+3 z^4+4 z^2-6 z^{-2} +7 }[/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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 12                     1 -1 10                   3   3 8                 5 1   -4 6               8 3     5 4             8 7       -1 2           10 6         4 0         7 12           5 -2       6 6             0 -4     3 9               6 -6   1 4                 -3 -8   3                   3 -10 1                     -1

Integral Khovanov Homology (db, data source)

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