L11n457

L11n456

L11n458

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Link Presentations

[edit Notes on L11n457's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X15,18,16,11 X20,9,21,10 X22,13,19,14 X14,21,15,22 X10,19,5,20 X17,8,18,9 X7,16,8,17 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {7, -4, 6, -5}, {10, -1, -9, 8, 4, -7}, {11, -2, 5, -6, -3, 9, -8, 3}

A Braid Representative	{{{braid_table}}}
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{t(1) t(4)^2 t(3)^2+t(2) t(4)^2 t(3)^2-t(4)^2 t(3)^2-t(1) t(4) t(3)^2+t(1) t(2) t(4) t(3)^2-t(2) t(4) t(3)^2+t(4) t(3)^2-t(1) t(4)^2 t(3)+t(1) t(2) t(4)^2 t(3)-t(2) t(4)^2 t(3)+t(4)^2 t(3)-t(1) t(3)+t(1) t(2) t(3)-t(2) t(3)+2 t(1) t(4) t(3)-3 t(1) t(2) t(4) t(3)+2 t(2) t(4) t(3)-3 t(4) t(3)+t(3)+t(1)-t(1) t(2)+t(2)-t(1) t(4)+t(1) t(2) t(4)-t(2) t(4)+t(4)}{\sqrt{t(1)} \sqrt{t(2)} t(3) t(4)} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{1}{q^{5/2}}+\frac{3}{q^{7/2}}-\frac{7}{q^{9/2}}+\frac{8}{q^{11/2}}-\frac{12}{q^{13/2}}+\frac{10}{q^{15/2}}-\frac{11}{q^{17/2}}+\frac{6}{q^{19/2}}-\frac{5}{q^{21/2}}+\frac{1}{q^{23/2}} }[/math] (db)
Signature	-5 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^{13} z^{-3} -2 z a^{11}-5 a^{11} z^{-1} -3 a^{11} z^{-3} +5 z^3 a^9+14 z a^9+10 a^9 z^{-1} +3 a^9 z^{-3} -3 z^5 a^7-11 z^3 a^7-12 z a^7-5 a^7 z^{-1} -a^7 z^{-3} -z^5 a^5-2 z^3 a^5 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^4 a^{14}-5 z^5 a^{13}+5 z^3 a^{13}+4 z a^{13}-5 a^{13} z^{-1} +a^{13} z^{-3} -z^8 a^{12}-2 z^6 a^{12}+3 z^4 a^{12}-6 z^2 a^{12}-3 a^{12} z^{-2} +10 a^{12}-z^9 a^{11}-z^7 a^{11}-z^5 a^{11}-3 z^3 a^{11}+15 z a^{11}-12 a^{11} z^{-1} +3 a^{11} z^{-3} -5 z^8 a^{10}+6 z^6 a^{10}+4 z^4 a^{10}-20 z^2 a^{10}-6 a^{10} z^{-2} +19 a^{10}-z^9 a^9-7 z^7 a^9+22 z^5 a^9-29 z^3 a^9+23 z a^9-12 a^9 z^{-1} +3 a^9 z^{-3} -4 z^8 a^8+5 z^6 a^8+5 z^4 a^8-14 z^2 a^8-3 a^8 z^{-2} +10 a^8-6 z^7 a^7+17 z^5 a^7-19 z^3 a^7+12 z a^7-5 a^7 z^{-1} +a^7 z^{-3} -3 z^6 a^6+5 z^4 a^6-z^5 a^5+2 z^3 a^5 }[/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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -4                   1 1 -6                 3 1 -2 -8               4     4 -10             4 3     -1 -12           8 4       4 -14         4 6         2 -16       7 6           1 -18     3 8             5 -20   2 3               -1 -22   4                 4 -24 1                   -1

Integral Khovanov Homology (db, data source)

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