L11a527

L11a526

L11a528

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Link Presentations

[edit Notes on L11a527's Link Presentations]

Planar diagram presentation	X8192 X14,3,15,4 X12,15,7,16 X10,21,11,22 X16,9,17,10 X22,11,13,12 X18,6,19,5 X20,18,21,17 X2738 X4,13,5,14 X6,20,1,19
Gauss code	{1, -9, 2, -10, 7, -11}, {9, -1, 5, -4, 6, -3}, {10, -2, 3, -5, 8, -7, 11, -8, 4, -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{u^2 v^2 w^2-u^2 v^2 w+u^2 v w^3-4 u^2 v w^2+4 u^2 v w-u^2 v+2 u^2 w^2-3 u^2 w+u^2+u v^2 w^3-3 u v^2 w^2+3 u v^2 w-2 u v w^3+7 u v w^2-7 u v w+2 u v-3 u w^2+3 u w-u-v^2 w^3+3 v^2 w^2-2 v^2 w+v w^3-4 v w^2+4 v w-v+w^2-w}{u v w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{-9} -3 q^{-8} +8 q^{-7} -13 q^{-6} +19 q^{-5} -21 q^{-4} +23 q^{-3} -q^2-19 q^{-2} +4 q+15 q^{-1} -9 }[/math] (db)
Signature	-2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^2 a^8+a^8 z^{-2} +2 a^8-3 z^4 a^6-8 z^2 a^6-2 a^6 z^{-2} -8 a^6+2 z^6 a^4+7 z^4 a^4+11 z^2 a^4+a^4 z^{-2} +7 a^4+z^6 a^2+z^4 a^2-z^2 a^2-a^2-z^4-z^2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{10} z^6-3 a^{10} z^4+3 a^{10} z^2-a^{10}+3 a^9 z^7-7 a^9 z^5+4 a^9 z^3+5 a^8 z^8-10 a^8 z^6+6 a^8 z^4-5 a^8 z^2-a^8 z^{-2} +4 a^8+5 a^7 z^9-6 a^7 z^7-3 a^7 z^5+6 a^7 z^3-6 a^7 z+2 a^7 z^{-1} +2 a^6 z^{10}+10 a^6 z^8-35 a^6 z^6+43 a^6 z^4-31 a^6 z^2-2 a^6 z^{-2} +12 a^6+12 a^5 z^9-21 a^5 z^7+8 a^5 z^5+7 a^5 z^3-8 a^5 z+2 a^5 z^{-1} +2 a^4 z^{10}+15 a^4 z^8-45 a^4 z^6+52 a^4 z^4-31 a^4 z^2-a^4 z^{-2} +10 a^4+7 a^3 z^9-4 a^3 z^7-9 a^3 z^5+11 a^3 z^3-3 a^3 z+10 a^2 z^8-17 a^2 z^6+13 a^2 z^4-7 a^2 z^2+2 a^2+8 a z^7-12 a z^5+z^5 a^{-1} +5 a z^3-z^3 a^{-1} -a z+4 z^6-5 z^4+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 5                       1 -1 3                     3   3 1                   6 1   -5 -1                 9 3     6 -3               11 7       -4 -5             12 8         4 -7           10 12           2 -9         9 11             -2 -11       5 11               6 -13     3 8                 -5 -15   1 6                   5 -17   2                     -2 -19 1                       1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-3 }[/math] [math]\displaystyle{ i=-1 }[/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}^{6}\oplus{\mathbb Z}_2^{2} }[/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}^{11}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/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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