L11a532

L11a531

L11a533

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Link Presentations

[edit Notes on L11a532's Link Presentations]

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

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 v w x^3-u v w x^2+u v w x-u v w-u v x^3+2 u v x^2-2 u v x+2 u v-2 u w x^3+2 u w x^2-2 u w x+u w+2 u x^3-3 u x^2+3 u x-2 u-2 v w x^3+3 v w x^2-3 v w x+2 v w+v x^3-2 v x^2+2 v x-2 v+2 w x^3-2 w x^2+2 w x-w-x^3+x^2-x+1}{\sqrt{u} \sqrt{v} \sqrt{w} x^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{11/2}-4 q^{9/2}+10 q^{7/2}-14 q^{5/2}+17 q^{3/2}-19 \sqrt{q}+\frac{16}{\sqrt{q}}-\frac{16}{q^{3/2}}+\frac{7}{q^{5/2}}-\frac{6}{q^{7/2}}+\frac{1}{q^{9/2}}-\frac{1}{q^{11/2}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^7 a^{-1} +3 a z^5-4 z^5 a^{-1} +z^5 a^{-3} -3 a^3 z^3+12 a z^3-9 z^3 a^{-1} +2 z^3 a^{-3} +a^5 z-11 a^3 z+22 a z-15 z a^{-1} +3 z a^{-3} +3 a^5 z^{-1} -14 a^3 z^{-1} +22 a z^{-1} -14 a^{-1} z^{-1} +3 a^{-3} z^{-1} +2 a^5 z^{-3} -7 a^3 z^{-3} +9 a z^{-3} -5 a^{-1} z^{-3} + a^{-3} z^{-3} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -a^2 z^{10}-z^{10}-a^3 z^9-6 a z^9-5 z^9 a^{-1} -a^4 z^8-a^2 z^8-11 z^8 a^{-2} -11 z^8-a^5 z^7-3 a^3 z^7+6 a z^7-6 z^7 a^{-1} -14 z^7 a^{-3} +a^4 z^6-2 a^2 z^6+9 z^6 a^{-2} -10 z^6 a^{-4} +16 z^6+6 a^5 z^5+23 a^3 z^5+21 a z^5+28 z^5 a^{-1} +20 z^5 a^{-3} -4 z^5 a^{-5} +10 a^4 z^4+36 a^2 z^4+19 z^4 a^{-2} +10 z^4 a^{-4} -z^4 a^{-6} +34 z^4-14 a^5 z^3-42 a^3 z^3-42 a z^3-22 z^3 a^{-1} -8 z^3 a^{-3} -26 a^4 z^2-73 a^2 z^2-34 z^2 a^{-2} -6 z^2 a^{-4} -75 z^2+16 a^5 z+39 a^3 z+37 a z+16 z a^{-1} +2 z a^{-3} +23 a^4+60 a^2+24 a^{-2} +4 a^{-4} +58-9 a^5 z^{-1} -23 a^3 z^{-1} -24 a z^{-1} -12 a^{-1} z^{-1} -2 a^{-3} z^{-1} -7 a^4 z^{-2} -19 a^2 z^{-2} -7 a^{-2} z^{-2} - a^{-4} z^{-2} -18 z^{-2} +2 a^5 z^{-3} +7 a^3 z^{-3} +9 a z^{-3} +5 a^{-1} z^{-3} + a^{-3} z^{-3} }[/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 12                       1 -1 10                     3   3 8                   7 1   -6 6                 7 3     4 4               10 7       -3 2             9 7         2 0           11 14           3 -2         5 5             0 -4       2 11               9 -6     4 5                 -1 -8   1 6                   5 -10                         0 -12 1                       1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=0 }[/math] [math]\displaystyle{ i=2 }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/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.

L11a531 L11a533