L11a353

L11a352

L11a354

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Link Presentations

[edit Notes on L11a353's Link Presentations]

Planar diagram presentation	X12,1,13,2 X8493 X16,6,17,5 X22,8,11,7 X20,15,21,16 X14,21,15,22 X6,14,7,13 X4,20,5,19 X18,9,19,10 X2,11,3,12 X10,17,1,18
Gauss code	{1, -10, 2, -8, 3, -7, 4, -2, 9, -11}, {10, -1, 7, -6, 5, -3, 11, -9, 8, -5, 6, -4}

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-1) (v-1) \left(u^2 v^2-3 u^2 v+2 u^2-u v^2+5 u v-u+2 v^2-3 v+1\right)}{u^{3/2} v^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{11/2}-4 q^{9/2}+10 q^{7/2}-17 q^{5/2}+21 q^{3/2}-25 \sqrt{q}+\frac{24}{\sqrt{q}}-\frac{21}{q^{3/2}}+\frac{15}{q^{5/2}}-\frac{9}{q^{7/2}}+\frac{4}{q^{9/2}}-\frac{1}{q^{11/2}} }[/math] (db)
Signature	1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^7 a^{-1} +3 a z^5-4 z^5 a^{-1} +z^5 a^{-3} -3 a^3 z^3+7 a z^3-9 z^3 a^{-1} +2 z^3 a^{-3} +a^5 z-3 a^3 z+6 a z-7 z a^{-1} +3 z a^{-3} +a z^{-1} - a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^4 a^{-6} +a^5 z^7-3 a^5 z^5+4 z^5 a^{-5} +3 a^5 z^3-a^5 z+4 a^4 z^8-13 a^4 z^6+10 z^6 a^{-4} +14 a^4 z^4-7 z^4 a^{-4} -5 a^4 z^2+3 z^2 a^{-4} +6 a^3 z^9-16 a^3 z^7+17 z^7 a^{-3} +9 a^3 z^5-25 z^5 a^{-3} +4 a^3 z^3+16 z^3 a^{-3} -2 a^3 z-5 z a^{-3} +3 a^2 z^{10}+7 a^2 z^8+18 z^8 a^{-2} -42 a^2 z^6-29 z^6 a^{-2} +43 a^2 z^4+12 z^4 a^{-2} -11 a^2 z^2-z^2 a^{-2} +17 a z^9+11 z^9 a^{-1} -38 a z^7-4 z^7 a^{-1} +9 a z^5-32 z^5 a^{-1} +14 a z^3+29 z^3 a^{-1} -6 a z-10 z a^{-1} +a z^{-1} + a^{-1} z^{-1} +3 z^{10}+21 z^8-68 z^6+49 z^4-10 z^2-1 }[/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 12                       1 -1 10                     3   3 8                   7 1   -6 6                 10 3     7 4               11 7       -4 2             14 10         4 0           12 13           1 -2         9 12             -3 -4       6 12               6 -6     3 9                 -6 -8   1 6                   5 -10   3                     -3 -12 1                       1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=0 }[/math] [math]\displaystyle{ i=2 }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{14} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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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