L11a145

L11a144

L11a146

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Link Presentations

[edit Notes on L11a145's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X18,12,19,11 X12,6,13,5 X4,19,5,20 X14,7,15,8 X20,13,21,14 X22,15,7,16 X16,21,17,22 X6,18,1,17
Gauss code	{1, -2, 3, -6, 5, -11}, {7, -1, 2, -3, 4, -5, 8, -7, 9, -10, 11, -4, 6, -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{2 t(1)^2 t(2)^4-2 t(1) t(2)^4-5 t(1)^2 t(2)^3+7 t(1) t(2)^3-2 t(2)^3+5 t(1)^2 t(2)^2-9 t(1) t(2)^2+5 t(2)^2-2 t(1)^2 t(2)+7 t(1) t(2)-5 t(2)-2 t(1)+2}{t(1) t(2)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{17}{q^{9/2}}-\frac{18}{q^{7/2}}+\frac{15}{q^{5/2}}+q^{3/2}-\frac{12}{q^{3/2}}-\frac{1}{q^{19/2}}+\frac{3}{q^{17/2}}-\frac{6}{q^{15/2}}+\frac{11}{q^{13/2}}-\frac{15}{q^{11/2}}-4 \sqrt{q}+\frac{7}{\sqrt{q}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^5 a^7+3 z^3 a^7+2 z a^7-z^7 a^5-4 z^5 a^5-6 z^3 a^5-4 z a^5-z^7 a^3-3 z^5 a^3-z^3 a^3+2 z a^3+a^3 z^{-1} +z^5 a+2 z^3 a-z a-a z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^5 a^{11}+2 z^3 a^{11}-3 z^6 a^{10}+6 z^4 a^{10}-2 z^2 a^{10}-5 z^7 a^9+10 z^5 a^9-6 z^3 a^9+z a^9-6 z^8 a^8+13 z^6 a^8-13 z^4 a^8+4 z^2 a^8-5 z^9 a^7+10 z^7 a^7-12 z^5 a^7+5 z^3 a^7-z a^7-2 z^{10} a^6-4 z^8 a^6+19 z^6 a^6-28 z^4 a^6+12 z^2 a^6-10 z^9 a^5+29 z^7 a^5-38 z^5 a^5+23 z^3 a^5-5 z a^5-2 z^{10} a^4-4 z^8 a^4+20 z^6 a^4-21 z^4 a^4+8 z^2 a^4-5 z^9 a^3+10 z^7 a^3-4 z^5 a^3+4 z^3 a^3-4 z a^3+a^3 z^{-1} -6 z^8 a^2+16 z^6 a^2-10 z^4 a^2+2 z^2 a^2-a^2-4 z^7 a+11 z^5 a-6 z^3 a-z a+a z^{-1} -z^6+2 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                     3   3 0                   4 1   -3 -2                 8 3     5 -4               8 5       -3 -6             10 7         3 -8           8 9           1 -10         7 9             -2 -12       4 8               4 -14     2 7                 -5 -16   1 4                   3 -18   2                     -2 -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}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/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=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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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