L11a547

L11a546

L11a548

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Link L11a547.

A graph, L11a547.

A part of a knot and a part of a graph.

Link Presentations

[edit Notes on L11a547's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X20,7,21,8 X10,19,5,20 X22,13,19,14 X18,21,11,22 X14,17,15,18 X16,9,17,10 X8,15,9,16 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {4, -3, 6, -5}, {10, -1, 3, -9, 8, -4}, {11, -2, 5, -7, 9, -8, 7, -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{t(1) t(4)^2 t(3)^2+2 t(2) t(4)^2 t(3)^2-2 t(4)^2 t(3)^2+t(1) t(3)^2-t(1) t(2) t(3)^2+t(2) t(3)^2-2 t(1) t(4) t(3)^2+2 t(1) t(2) t(4) t(3)^2-3 t(2) t(4) t(3)^2+3 t(4) t(3)^2-t(3)^2-2 t(1) t(4)^2 t(3)+2 t(1) t(2) t(4)^2 t(3)-3 t(2) t(4)^2 t(3)+3 t(4)^2 t(3)-3 t(1) t(3)+3 t(1) t(2) t(3)-2 t(2) t(3)+5 t(1) t(4) t(3)-6 t(1) t(2) t(4) t(3)+5 t(2) t(4) t(3)-6 t(4) t(3)+2 t(3)+t(1) t(4)^2-t(1) t(2) t(4)^2+t(2) t(4)^2-t(4)^2+2 t(1)-2 t(1) t(2)+t(2)-3 t(1) t(4)+3 t(1) t(2) t(4)-2 t(2) t(4)+2 t(4)}{\sqrt{t(1)} \sqrt{t(2)} t(3) t(4)} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{11}{q^{9/2}}+\frac{4}{q^{7/2}}-\frac{1}{q^{5/2}}+\frac{1}{q^{27/2}}-\frac{4}{q^{25/2}}+\frac{9}{q^{23/2}}-\frac{17}{q^{21/2}}+\frac{21}{q^{19/2}}-\frac{27}{q^{17/2}}+\frac{25}{q^{15/2}}-\frac{24}{q^{13/2}}+\frac{16}{q^{11/2}} }[/math] (db)
Signature	-5 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z a^{13}+a^{13} z^{-3} +3 z^3 a^{11}+z a^{11}-5 a^{11} z^{-1} -3 a^{11} z^{-3} -2 z^5 a^9+z^3 a^9+11 z a^9+10 a^9 z^{-1} +3 a^9 z^{-3} -4 z^5 a^7-11 z^3 a^7-11 z a^7-5 a^7 z^{-1} -a^7 z^{-3} -z^5 a^5-z^3 a^5 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{16} z^6-2 a^{16} z^4+a^{16} z^2+4 a^{15} z^7-9 a^{15} z^5+8 a^{15} z^3-3 a^{15} z+7 a^{14} z^8-13 a^{14} z^6+6 a^{14} z^4+a^{14} z^2+6 a^{13} z^9-23 a^{13} z^5+25 a^{13} z^3-a^{13} z^{-3} -12 a^{13} z+5 a^{13} z^{-1} +2 a^{12} z^{10}+18 a^{12} z^8-45 a^{12} z^6+26 a^{12} z^4+3 a^{12} z^2+3 a^{12} z^{-2} -10 a^{12}+14 a^{11} z^9-10 a^{11} z^7-26 a^{11} z^5+31 a^{11} z^3-3 a^{11} z^{-3} -21 a^{11} z+12 a^{11} z^{-1} +2 a^{10} z^{10}+23 a^{10} z^8-50 a^{10} z^6+22 a^{10} z^4+15 a^{10} z^2+6 a^{10} z^{-2} -19 a^{10}+8 a^9 z^9+4 a^9 z^7-30 a^9 z^5+32 a^9 z^3-3 a^9 z^{-3} -23 a^9 z+12 a^9 z^{-1} +12 a^8 z^8-15 a^8 z^6+a^8 z^4+12 a^8 z^2+3 a^8 z^{-2} -10 a^8+10 a^7 z^7-17 a^7 z^5+17 a^7 z^3-a^7 z^{-3} -11 a^7 z+5 a^7 z^{-1} +4 a^6 z^6-3 a^6 z^4+a^5 z^5-a^5 z^3 }[/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   \ -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -4                       1 1 -6                     4 1 -3 -8                   7     7 -10                 9 4     -5 -12               15 7       8 -14             12 11         -1 -16           15 13           2 -18         10 16             6 -20       7 11               -4 -22     3 11                 8 -24   1 6                   -5 -26   3                     3 -28 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=-11 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-10 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-9 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{16}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{15} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{13} }[/math] [math]\displaystyle{ {\mathbb Z}^{15} }[/math] [math]\displaystyle{ r=-3 }[/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}^{4}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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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