L10a111

L10a110

L10a112

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Link Presentations

[edit Notes on L10a111's Link Presentations]

Planar diagram presentation	X12,1,13,2 X16,7,17,8 X8394 X2,18,3,17 X14,6,15,5 X6,12,7,11 X18,10,19,9 X20,15,11,16 X10,13,1,14 X4,19,5,20
Gauss code	{1, -4, 3, -10, 5, -6, 2, -3, 7, -9}, {6, -1, 9, -5, 8, -2, 4, -7, 10, -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{(t(1)-1) (t(2)-1) \left(t(2)^2 t(1)^2-2 t(2) t(1)^2+t(1)^2-2 t(2)^2 t(1)+3 t(2) t(1)-2 t(1)+t(2)^2-2 t(2)+1\right)}{t(1)^{3/2} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{9}{q^{9/2}}-q^{7/2}+\frac{15}{q^{7/2}}+5 q^{5/2}-\frac{19}{q^{5/2}}-11 q^{3/2}+\frac{20}{q^{3/2}}-\frac{1}{q^{13/2}}+\frac{4}{q^{11/2}}+15 \sqrt{q}-\frac{20}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a z^7-2 a^3 z^5+3 a z^5-z^5 a^{-1} +a^5 z^3-4 a^3 z^3+3 a z^3-z^3 a^{-1} +a^5 z-2 a^3 z+a z+a z^{-1} - a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -4 a^3 z^9-4 a z^9-9 a^4 z^8-20 a^2 z^8-11 z^8-8 a^5 z^7-12 a^3 z^7-15 a z^7-11 z^7 a^{-1} -4 a^6 z^6+11 a^4 z^6+34 a^2 z^6-5 z^6 a^{-2} +14 z^6-a^7 z^5+12 a^5 z^5+34 a^3 z^5+39 a z^5+17 z^5 a^{-1} -z^5 a^{-3} +5 a^6 z^4-3 a^4 z^4-14 a^2 z^4+4 z^4 a^{-2} -2 z^4+a^7 z^3-8 a^5 z^3-22 a^3 z^3-19 a z^3-6 z^3 a^{-1} -2 a^6 z^2-a^4 z^2+2 a^2 z^2+z^2+2 a^5 z+4 a^3 z+2 a z+1-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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 8                     1 1 6                   4   -4 4                 7 1   6 2               8 4     -4 0             12 7       5 -2           10 10         0 -4         9 10           -1 -6       6 10             4 -8     3 9               -6 -10   1 6                 5 -12   3                   -3 -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}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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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