L10a1

L9n28

L10a2

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Link Presentations

[edit Notes on L10a1's Link Presentations]

Planar diagram presentation	X6172 X14,7,15,8 X4,15,1,16 X10,6,11,5 X8493 X18,11,19,12 X20,17,5,18 X12,19,13,20 X16,10,17,9 X2,14,3,13
Gauss code	{1, -10, 5, -3}, {4, -1, 2, -5, 9, -4, 6, -8, 10, -2, 3, -9, 7, -6, 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-1) (v-1) \left(v^4-3 v^3+5 v^2-3 v+1\right)}{\sqrt{u} v^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{9}{q^{9/2}}-q^{7/2}+\frac{13}{q^{7/2}}+4 q^{5/2}-\frac{17}{q^{5/2}}-8 q^{3/2}+\frac{17}{q^{3/2}}-\frac{1}{q^{13/2}}+\frac{5}{q^{11/2}}+12 \sqrt{q}-\frac{17}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a z^7-2 a^3 z^5+4 a z^5-z^5 a^{-1} +a^5 z^3-4 a^3 z^3+6 a z^3-2 z^3 a^{-1} +a z-z a^{-1} -a^5 z^{-1} +3 a^3 z^{-1} -2 a z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -2 a^3 z^9-2 a z^9-7 a^4 z^8-13 a^2 z^8-6 z^8-9 a^5 z^7-17 a^3 z^7-15 a z^7-7 z^7 a^{-1} -5 a^6 z^6+3 a^4 z^6+15 a^2 z^6-4 z^6 a^{-2} +3 z^6-a^7 z^5+15 a^5 z^5+40 a^3 z^5+37 a z^5+12 z^5 a^{-1} -z^5 a^{-3} +6 a^6 z^4+10 a^4 z^4+5 a^2 z^4+6 z^4 a^{-2} +7 z^4-6 a^5 z^3-23 a^3 z^3-26 a z^3-8 z^3 a^{-1} +z^3 a^{-3} -2 a^4 z^2-5 a^2 z^2-2 z^2 a^{-2} -5 z^2+a^3 z+3 a z+2 z a^{-1} -a^6-3 a^4-3 a^2+a^5 z^{-1} +3 a^3 z^{-1} +2 a 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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 8                     1 1 6                   3   -3 4                 5 1   4 2               7 3     -4 0             10 5       5 -2           9 9         0 -4         8 8           0 -6       5 9             4 -8     4 8               -4 -10   1 5                 4 -12   4                   -4 -14 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=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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.

L9n28 L10a2