L11a149

L11a148

L11a150

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Link Presentations

[edit Notes on L11a149's Link Presentations]

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

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 u^2 v^4-5 u^2 v^3+5 u^2 v^2-2 u^2 v-2 u v^4+8 u v^3-11 u v^2+8 u v-2 u-2 v^3+5 v^2-5 v+2}{u v^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{19}{q^{9/2}}-\frac{19}{q^{7/2}}+\frac{15}{q^{5/2}}+q^{3/2}-\frac{12}{q^{3/2}}-\frac{1}{q^{19/2}}+\frac{4}{q^{17/2}}-\frac{8}{q^{15/2}}+\frac{13}{q^{13/2}}-\frac{17}{q^{11/2}}-3 \sqrt{q}+\frac{6}{\sqrt{q}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^7 z^5+2 a^7 z^3-a^7 z^{-1} -a^5 z^7-3 a^5 z^5-a^5 z^3+4 a^5 z+3 a^5 z^{-1} -a^3 z^7-4 a^3 z^5-7 a^3 z^3-7 a^3 z-2 a^3 z^{-1} +a z^5+3 a z^3+2 a z }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{11} z^5-a^{11} z^3+4 a^{10} z^6-6 a^{10} z^4+a^{10} z^2+7 a^9 z^7-11 a^9 z^5+3 a^9 z^3+8 a^8 z^8-13 a^8 z^6+6 a^8 z^4-a^8+6 a^7 z^9-8 a^7 z^7+5 a^7 z^5-2 a^7 z^3+a^7 z^{-1} +2 a^6 z^{10}+7 a^6 z^8-21 a^6 z^6+18 a^6 z^4-3 a^6+10 a^5 z^9-24 a^5 z^7+24 a^5 z^5-6 a^5 z^3-5 a^5 z+3 a^5 z^{-1} +2 a^4 z^{10}+3 a^4 z^8-13 a^4 z^6+10 a^4 z^4+a^4 z^2-3 a^4+4 a^3 z^9-6 a^3 z^7-2 a^3 z^5+8 a^3 z^3-8 a^3 z+2 a^3 z^{-1} +4 a^2 z^8-8 a^2 z^6+a^2 z^4+2 a^2 z^2+3 a z^7-9 a z^5+8 a z^3-3 a z+z^6-3 z^4+2 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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 4                       1 -1 2                     2   2 0                   4 1   -3 -2                 8 2     6 -4               8 5       -3 -6             11 7         4 -8           9 9           0 -10         8 10             -2 -12       5 9               4 -14     3 8                 -5 -16   1 5                   4 -18   3                     -3 -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}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/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}^{2}\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^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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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