L11a115

L11a114

L11a116

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Link Presentations

[edit Notes on L11a115's Link Presentations]

Planar diagram presentation	X6172 X14,3,15,4 X16,8,17,7 X22,18,5,17 X18,11,19,12 X20,9,21,10 X10,19,11,20 X8,21,9,22 X12,16,13,15 X2536 X4,13,1,14
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -8, 6, -7, 5, -9, 11, -2, 9, -3, 4, -5, 7, -6, 8, -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 v^3-7 u v^2+10 u v-4 u-4 v^3+10 v^2-7 v+2}{\sqrt{u} v^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{11}{q^{9/2}}+\frac{14}{q^{7/2}}+q^{5/2}-\frac{15}{q^{5/2}}-4 q^{3/2}+\frac{13}{q^{3/2}}-\frac{1}{q^{17/2}}+\frac{2}{q^{15/2}}-\frac{5}{q^{13/2}}+\frac{8}{q^{11/2}}+7 \sqrt{q}-\frac{11}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^9 z^{-1} -3 a^7 z-2 a^7 z^{-1} +3 a^5 z^3+3 a^5 z+a^5 z^{-1} -a^3 z^5+a^3 z^{-1} -a z^5-a z^3+z^3 a^{-1} -2 a z-a z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^9 z^7-5 a^9 z^5+8 a^9 z^3-5 a^9 z+a^9 z^{-1} +2 a^8 z^8-8 a^8 z^6+9 a^8 z^4-4 a^8 z^2+a^8+2 a^7 z^9-4 a^7 z^7-7 a^7 z^5+18 a^7 z^3-12 a^7 z+2 a^7 z^{-1} +a^6 z^{10}+3 a^6 z^8-20 a^6 z^6+25 a^6 z^4-12 a^6 z^2+3 a^6+6 a^5 z^9-15 a^5 z^7+4 a^5 z^5+11 a^5 z^3-8 a^5 z+a^5 z^{-1} +a^4 z^{10}+8 a^4 z^8-29 a^4 z^6+30 a^4 z^4-12 a^4 z^2+2 a^4+4 a^3 z^9-2 a^3 z^7-7 a^3 z^5+4 a^3 z^3+2 a^3 z-a^3 z^{-1} +7 a^2 z^8-10 a^2 z^6+6 a^2 z^4+z^4 a^{-2} -4 a^2 z^2+a^2+8 a z^7-9 a z^5+4 z^5 a^{-1} -3 z^3 a^{-1} +3 a z-a z^{-1} +7 z^6-7 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 6                       1 -1 4                     3   3 2                   4 1   -3 0                 7 3     4 -2               7 5       -2 -4             8 6         2 -6           6 7           1 -8         5 8             -3 -10       4 7               3 -12     1 4                 -3 -14   1 4                   3 -16   1                     -1 -18 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=-8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/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}^{5}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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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