L11a501

L11a500

L11a502

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Link Presentations

[edit Notes on L11a501's Link Presentations]

Planar diagram presentation	X8192 X14,5,15,6 X10,3,11,4 X4,13,5,14 X2738 X6,9,1,10 X18,12,19,11 X12,18,7,17 X20,16,21,15 X22,20,13,19 X16,22,17,21
Gauss code	{1, -5, 3, -4, 2, -6}, {5, -1, 6, -3, 7, -8}, {4, -2, 9, -11, 8, -7, 10, -9, 11, -10}

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 w^3-3 u^2 v w^2+3 u^2 v w-u^2 v-u^2 w^3+2 u^2 w^2-2 u^2 w+u v^2 w^3-3 u v^2 w^2+3 u v^2 w-u v^2-2 u v w^3+6 u v w^2-6 u v w+2 u v+u w^3-3 u w^2+3 u w-u+2 v^2 w^2-2 v^2 w+v^2+v w^3-3 v w^2+3 v w-v}{u v w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^5+3 q^4-7 q^3+12 q^2-16 q+19-18 q^{-1} +17 q^{-2} -11 q^{-3} +8 q^{-4} -3 q^{-5} + q^{-6} }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^4 z^4+2 a^4 z^2-z^2 a^{-4} +2 a^4 z^{-2} +3 a^4- a^{-4} -a^2 z^6-3 a^2 z^4+2 z^4 a^{-2} -7 a^2 z^2+3 z^2 a^{-2} -5 a^2 z^{-2} - a^{-2} z^{-2} -9 a^2+ a^{-2} -z^6-z^4+2 z^2+4 z^{-2} +6 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^2 z^{10}+z^{10}+4 a^3 z^9+7 a z^9+3 z^9 a^{-1} +6 a^4 z^8+12 a^2 z^8+5 z^8 a^{-2} +11 z^8+3 a^5 z^7-2 a^3 z^7-2 a z^7+8 z^7 a^{-1} +5 z^7 a^{-3} +a^6 z^6-19 a^4 z^6-39 a^2 z^6-2 z^6 a^{-2} +3 z^6 a^{-4} -24 z^6-7 a^5 z^5-17 a^3 z^5-29 a z^5-27 z^5 a^{-1} -7 z^5 a^{-3} +z^5 a^{-5} -3 a^6 z^4+26 a^4 z^4+51 a^2 z^4-6 z^4 a^{-2} -5 z^4 a^{-4} +21 z^4+3 a^5 z^3+28 a^3 z^3+50 a z^3+31 z^3 a^{-1} +4 z^3 a^{-3} -2 z^3 a^{-5} +2 a^6 z^2-23 a^4 z^2-43 a^2 z^2+5 z^2 a^{-2} +3 z^2 a^{-4} -16 z^2-19 a^3 z-35 a z-19 z a^{-1} -2 z a^{-3} +z a^{-5} +11 a^4+22 a^2- a^{-4} +13+5 a^3 z^{-1} +9 a z^{-1} +5 a^{-1} z^{-1} + a^{-3} z^{-1} -2 a^4 z^{-2} -5 a^2 z^{-2} - a^{-2} z^{-2} -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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 11                       1 -1 9                     2   2 7                   5 1   -4 5                 7 2     5 3               9 5       -4 1             10 7         3 -1           10 11           1 -3         7 8             -1 -5       5 11               6 -7     3 6                 -3 -9     5                   5 -11 1 3                     -2 -13 1                       1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-1 }[/math] [math]\displaystyle{ i=1 }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=5 }[/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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