L11a91

L11a90

L11a92

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Link Presentations

[edit Notes on L11a91's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X18,8,19,7 X22,20,5,19 X20,9,21,10 X8,21,9,22 X16,11,17,12 X14,17,15,18 X10,15,11,16 X2536 X4,14,1,13
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -6, 5, -9, 7, -2, 11, -8, 9, -7, 8, -3, 4, -5, 6, -4}

A Braid Representative	{{{braid_table}}}
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^4-4 v^3+7 v^2-4 v+1\right)}{\sqrt{u} v^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^{7/2}+4 q^{5/2}-10 q^{3/2}+15 \sqrt{q}-\frac{20}{\sqrt{q}}+\frac{22}{q^{3/2}}-\frac{22}{q^{5/2}}+\frac{18}{q^{7/2}}-\frac{13}{q^{9/2}}+\frac{7}{q^{11/2}}-\frac{3}{q^{13/2}}+\frac{1}{q^{15/2}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z a^7-a^7 z^{-1} +3 z^3 a^5+6 z a^5+4 a^5 z^{-1} -3 z^5 a^3-9 z^3 a^3-12 z a^3-6 a^3 z^{-1} +z^7 a+4 z^5 a+9 z^3 a+10 z a+5 a z^{-1} -z^5 a^{-1} -2 z^3 a^{-1} -3 z a^{-1} -2 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -a^4 z^{10}-a^2 z^{10}-3 a^5 z^9-8 a^3 z^9-5 a z^9-4 a^6 z^8-13 a^4 z^8-19 a^2 z^8-10 z^8-3 a^7 z^7-6 a^5 z^7-6 a^3 z^7-12 a z^7-9 z^7 a^{-1} -a^8 z^6+5 a^6 z^6+27 a^4 z^6+38 a^2 z^6-4 z^6 a^{-2} +13 z^6+8 a^7 z^5+29 a^5 z^5+50 a^3 z^5+45 a z^5+15 z^5 a^{-1} -z^5 a^{-3} +3 a^8 z^4+4 a^6 z^4-10 a^4 z^4-19 a^2 z^4+4 z^4 a^{-2} -4 z^4-8 a^7 z^3-34 a^5 z^3-60 a^3 z^3-47 a z^3-12 z^3 a^{-1} +z^3 a^{-3} -3 a^8 z^2-8 a^6 z^2-6 a^4 z^2-z^2 a^{-2} +4 a^7 z+18 a^5 z+31 a^3 z+24 a z+7 z a^{-1} +a^8+3 a^6+3 a^4+a^2+1-a^7 z^{-1} -4 a^5 z^{-1} -6 a^3 z^{-1} -5 a z^{-1} -2 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                     3   -3 4                   7 1   6 2                 8 3     -5 0               12 7       5 -2             12 10         -2 -4           10 10           0 -6         8 12             4 -8       5 10               -5 -10     2 8                 6 -12   1 5                   -4 -14   2                     2 -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}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/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}^{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}^{3}\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^{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.

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