L11a337

L11a336

L11a338

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Link Presentations

[edit Notes on L11a337's Link Presentations]

Planar diagram presentation	X10,1,11,2 X22,5,9,6 X14,3,15,4 X4,15,5,16 X16,7,17,8 X20,17,21,18 X18,11,19,12 X12,19,13,20 X2,9,3,10 X8,13,1,14 X6,21,7,22
Gauss code	{1, -9, 3, -4, 2, -11, 5, -10}, {9, -1, 7, -8, 10, -3, 4, -5, 6, -7, 8, -6, 11, -2}

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(2)^5-4 t(1)^2 t(2)^4+5 t(1) t(2)^4-t(2)^4-2 t(1)^3 t(2)^3+9 t(1)^2 t(2)^3-10 t(1) t(2)^3+2 t(2)^3+2 t(1)^3 t(2)^2-10 t(1)^2 t(2)^2+9 t(1) t(2)^2-2 t(2)^2-t(1)^3 t(2)+5 t(1)^2 t(2)-4 t(1) t(2)-t(1)^2}{t(1)^{3/2} t(2)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{8}{q^{9/2}}+\frac{3}{q^{7/2}}-\frac{1}{q^{5/2}}+\frac{1}{q^{27/2}}-\frac{4}{q^{25/2}}+\frac{9}{q^{23/2}}-\frac{15}{q^{21/2}}+\frac{19}{q^{19/2}}-\frac{22}{q^{17/2}}+\frac{22}{q^{15/2}}-\frac{19}{q^{13/2}}+\frac{13}{q^{11/2}} }[/math] (db)
Signature	-5 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z a^{13}-a^{13} z^{-1} +4 z^3 a^{11}+7 z a^{11}+3 a^{11} z^{-1} -3 z^5 a^9-7 z^3 a^9-5 z a^9-2 a^9 z^{-1} -3 z^5 a^7-7 z^3 a^7-4 z a^7-z^5 a^5-2 z^3 a^5-z a^5 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^6 a^{16}+2 z^4 a^{16}-z^2 a^{16}-4 z^7 a^{15}+9 z^5 a^{15}-6 z^3 a^{15}+z a^{15}-7 z^8 a^{14}+15 z^6 a^{14}-8 z^4 a^{14}+2 z^2 a^{14}-a^{14}-6 z^9 a^{13}+6 z^7 a^{13}+8 z^5 a^{13}-6 z^3 a^{13}+a^{13} z^{-1} -2 z^{10} a^{12}-12 z^8 a^{12}+36 z^6 a^{12}-31 z^4 a^{12}+15 z^2 a^{12}-3 a^{12}-12 z^9 a^{11}+21 z^7 a^{11}-16 z^5 a^{11}+19 z^3 a^{11}-14 z a^{11}+3 a^{11} z^{-1} -2 z^{10} a^{10}-12 z^8 a^{10}+31 z^6 a^{10}-28 z^4 a^{10}+12 z^2 a^{10}-3 a^{10}-6 z^9 a^9+5 z^7 a^9-4 z^5 a^9+8 z^3 a^9-8 z a^9+2 a^9 z^{-1} -7 z^8 a^8+8 z^6 a^8-3 z^4 a^8-z^2 a^8-6 z^7 a^7+10 z^5 a^7-9 z^3 a^7+4 z a^7-3 z^6 a^6+4 z^4 a^6-z^2 a^6-z^5 a^5+2 z^3 a^5-z 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   \ -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -4                       1 1 -6                     3 1 -2 -8                   5     5 -10                 8 3     -5 -12               11 5       6 -14             11 8         -3 -16           11 11           0 -18         8 11             3 -20       7 11               -4 -22     3 9                 6 -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}^{9}\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^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/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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