L11a171

L11a170

L11a172

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Link Presentations

[edit Notes on L11a171's Link Presentations]

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

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 t(1)^2 t(2)^4-3 t(1) t(2)^4+t(2)^4-7 t(1)^2 t(2)^3+12 t(1) t(2)^3-5 t(2)^3+9 t(1)^2 t(2)^2-17 t(1) t(2)^2+9 t(2)^2-5 t(1)^2 t(2)+12 t(1) t(2)-7 t(2)+t(1)^2-3 t(1)+2}{t(1) t(2)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{3/2}-5 \sqrt{q}+\frac{11}{\sqrt{q}}-\frac{20}{q^{3/2}}+\frac{26}{q^{5/2}}-\frac{31}{q^{7/2}}+\frac{31}{q^{9/2}}-\frac{27}{q^{11/2}}+\frac{20}{q^{13/2}}-\frac{12}{q^{15/2}}+\frac{5}{q^{17/2}}-\frac{1}{q^{19/2}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^7 z^5+a^7 z^3-a^5 z^7-2 a^5 z^5-a^5 z^3-a^3 z^7-2 a^3 z^5-a^3 z^3+a^3 z^{-1} +a z^5+a z^3-a z-a z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{11} z^5+5 a^{10} z^6-3 a^{10} z^4+12 a^9 z^7-14 a^9 z^5+5 a^9 z^3+17 a^8 z^8-25 a^8 z^6+13 a^8 z^4-2 a^8 z^2+13 a^7 z^9-7 a^7 z^7-14 a^7 z^5+8 a^7 z^3+4 a^6 z^{10}+26 a^6 z^8-64 a^6 z^6+38 a^6 z^4-6 a^6 z^2+23 a^5 z^9-34 a^5 z^7+a^5 z^5+6 a^5 z^3+4 a^4 z^{10}+19 a^4 z^8-54 a^4 z^6+34 a^4 z^4-6 a^4 z^2+10 a^3 z^9-10 a^3 z^7-9 a^3 z^5+8 a^3 z^3+a^3 z-a^3 z^{-1} +10 a^2 z^8-19 a^2 z^6+11 a^2 z^4-2 a^2 z^2+a^2+5 a z^7-9 a z^5+5 a z^3+a z-a z^{-1} +z^6-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 4                       1 -1 2                     4   4 0                   7 1   -6 -2                 13 4     9 -4               14 8       -6 -6             17 12         5 -8           15 15           0 -10         12 16             -4 -12       8 15               7 -14     4 12                 -8 -16   1 8                   7 -18   4                     -4 -20 1                       1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-4 }[/math] [math]\displaystyle{ i=-2 }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{15}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{16}\oplus{\mathbb Z}_2^{15} }[/math] [math]\displaystyle{ {\mathbb Z}^{15} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{15}\oplus{\mathbb Z}_2^{16} }[/math] [math]\displaystyle{ {\mathbb Z}^{17} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{14} }[/math] [math]\displaystyle{ {\mathbb Z}^{14} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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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