L11a77

L11a76

L11a78

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Link Presentations

[edit Notes on L11a77's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X14,12,15,11 X22,15,5,16 X16,7,17,8 X20,17,21,18 X18,10,19,9 X8,20,9,19 X10,21,11,22 X2536 X4,14,1,13
Gauss code	{1, -10, 2, -11}, {10, -1, 5, -8, 7, -9, 3, -2, 11, -3, 4, -5, 6, -7, 8, -6, 9, -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^2-3 v+1\right) \left(v^2-v+1\right)}{\sqrt{u} v^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{9/2}-4 q^{7/2}+8 q^{5/2}-13 q^{3/2}+16 \sqrt{q}-\frac{20}{\sqrt{q}}+\frac{19}{q^{3/2}}-\frac{16}{q^{5/2}}+\frac{12}{q^{7/2}}-\frac{7}{q^{9/2}}+\frac{3}{q^{11/2}}-\frac{1}{q^{13/2}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^5 z^3+2 a^5 z+a^5 z^{-1} -2 a^3 z^5-6 a^3 z^3+z^3 a^{-3} -7 a^3 z+z a^{-3} -3 a^3 z^{-1} +a z^7+4 a z^5-2 z^5 a^{-1} +8 a z^3-5 z^3 a^{-1} +8 a z-4 z a^{-1} +4 a z^{-1} -2 a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -a^2 z^{10}-z^{10}-3 a^3 z^9-7 a z^9-4 z^9 a^{-1} -5 a^4 z^8-11 a^2 z^8-6 z^8 a^{-2} -12 z^8-5 a^5 z^7-8 a^3 z^7-z^7 a^{-1} -4 z^7 a^{-3} -3 a^6 z^6+2 a^4 z^6+20 a^2 z^6+13 z^6 a^{-2} -z^6 a^{-4} +29 z^6-a^7 z^5+7 a^5 z^5+25 a^3 z^5+29 a z^5+22 z^5 a^{-1} +10 z^5 a^{-3} +5 a^6 z^4+6 a^4 z^4-7 a^2 z^4-5 z^4 a^{-2} +2 z^4 a^{-4} -15 z^4+2 a^7 z^3-4 a^5 z^3-26 a^3 z^3-36 a z^3-23 z^3 a^{-1} -7 z^3 a^{-3} -3 a^6 z^2-7 a^4 z^2-5 a^2 z^2-z^2 a^{-2} -z^2 a^{-4} -z^2-a^7 z+2 a^5 z+13 a^3 z+17 a z+9 z a^{-1} +2 z a^{-3} +a^6+3 a^4+3 a^2+2-a^5 z^{-1} -3 a^3 z^{-1} -4 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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 10                       1 -1 8                     3   3 6                   5 1   -4 4                 8 3     5 2               8 5       -3 0             12 8         4 -2           9 10           1 -4         7 10             -3 -6       5 9               4 -8     2 7                 -5 -10   1 5                   4 -12   2                     -2 -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}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\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^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/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}^{8}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8} }[/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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