L11a180

L11a179

L11a181

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Link Presentations

[edit Notes on L11a180's Link Presentations]

Planar diagram presentation	X8192 X10,4,11,3 X14,6,15,5 X16,8,17,7 X22,18,7,17 X12,15,13,16 X20,10,21,9 X18,11,19,12 X6,14,1,13 X4,20,5,19 X2,21,3,22
Gauss code	{1, -11, 2, -10, 3, -9}, {4, -1, 7, -2, 8, -6, 9, -3, 6, -4, 5, -8, 10, -7, 11, -5}

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)^2 t(2)^4-2 t(1) t(2)^4+t(2)^4-5 t(1)^2 t(2)^3+11 t(1) t(2)^3-6 t(2)^3+9 t(1)^2 t(2)^2-19 t(1) t(2)^2+9 t(2)^2-6 t(1)^2 t(2)+11 t(1) t(2)-5 t(2)+t(1)^2-2 t(1)+1}{t(1) t(2)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{15/2}-4 q^{13/2}+10 q^{11/2}-18 q^{9/2}+24 q^{7/2}-29 q^{5/2}+29 q^{3/2}-26 \sqrt{q}+\frac{19}{\sqrt{q}}-\frac{12}{q^{3/2}}+\frac{5}{q^{5/2}}-\frac{1}{q^{7/2}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^7 a^{-1} +a z^5-3 z^5 a^{-1} +3 z^5 a^{-3} +a z^3-5 z^3 a^{-1} +5 z^3 a^{-3} -3 z^3 a^{-5} +a z-z a^{-1} +2 z a^{-3} -2 z a^{-5} +z a^{-7} + a^{-1} z^{-1} - a^{-3} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^6 a^{-8} -2 z^4 a^{-8} +z^2 a^{-8} +4 z^7 a^{-7} -8 z^5 a^{-7} +6 z^3 a^{-7} -2 z a^{-7} +8 z^8 a^{-6} -16 z^6 a^{-6} +12 z^4 a^{-6} -4 z^2 a^{-6} +8 z^9 a^{-5} -7 z^7 a^{-5} -10 z^5 a^{-5} +12 z^3 a^{-5} -3 z a^{-5} +3 z^{10} a^{-4} +19 z^8 a^{-4} -54 z^6 a^{-4} +41 z^4 a^{-4} -10 z^2 a^{-4} +19 z^9 a^{-3} -23 z^7 a^{-3} +a^3 z^5-12 z^5 a^{-3} +15 z^3 a^{-3} - a^{-3} z^{-1} +3 z^{10} a^{-2} +27 z^8 a^{-2} +5 a^2 z^6-65 z^6 a^{-2} -3 a^2 z^4+41 z^4 a^{-2} -8 z^2 a^{-2} + a^{-2} +11 z^9 a^{-1} +12 a z^7-15 a z^5-26 z^5 a^{-1} +6 a z^3+15 z^3 a^{-1} -a z- a^{-1} z^{-1} +16 z^8-23 z^6+11 z^4-3 z^2 }[/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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 16                       1 -1 14                     3   3 12                   7 1   -6 10                 11 3     8 8               13 7       -6 6             16 11         5 4           14 14           0 2         12 15             -3 0       8 15               7 -2     4 11                 -7 -4   1 8                   7 -6   4                     -4 -8 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=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{15}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{15}\oplus{\mathbb Z}_2^{14} }[/math] [math]\displaystyle{ {\mathbb Z}^{14} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{15} }[/math] [math]\displaystyle{ {\mathbb Z}^{16} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{13} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=7 }[/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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