L11a23

L11a22

L11a24

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Link Presentations

[edit Notes on L11a23's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X16,11,17,12 X14,7,15,8 X8,15,9,16 X20,13,21,14 X22,17,5,18 X18,21,19,22 X12,19,13,20 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -5, 11, -2, 3, -9, 6, -4, 5, -3, 7, -8, 9, -6, 8, -7}

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{2 t(2)^5+4 t(1) t(2)^4-6 t(2)^4-9 t(1) t(2)^3+10 t(2)^3+10 t(1) t(2)^2-9 t(2)^2-6 t(1) t(2)+4 t(2)+2 t(1)}{\sqrt{t(1)} 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{3}{q^{25/2}}+\frac{8}{q^{23/2}}-\frac{13}{q^{21/2}}+\frac{17}{q^{19/2}}-\frac{20}{q^{17/2}}+\frac{20}{q^{15/2}}-\frac{18}{q^{13/2}}+\frac{12}{q^{11/2}} }[/math] (db)
Signature	-5 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^{13} (-z)-a^{13} z^{-1} +3 a^{11} z^3+4 a^{11} z+a^{11} z^{-1} -2 a^9 z^5-2 a^9 z^3+3 a^9 z+2 a^9 z^{-1} -3 a^7 z^5-8 a^7 z^3-7 a^7 z-2 a^7 z^{-1} -a^5 z^5-2 a^5 z^3-a^5 z }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^6 a^{16}+3 z^4 a^{16}-3 z^2 a^{16}+a^{16}-3 z^7 a^{15}+7 z^5 a^{15}-5 z^3 a^{15}+z a^{15}-5 z^8 a^{14}+10 z^6 a^{14}-5 z^4 a^{14}+z^2 a^{14}-4 z^9 a^{13}+z^7 a^{13}+11 z^5 a^{13}-6 z^3 a^{13}-z a^{13}+a^{13} z^{-1} -z^{10} a^{12}-13 z^8 a^{12}+34 z^6 a^{12}-29 z^4 a^{12}+14 z^2 a^{12}-3 a^{12}-8 z^9 a^{11}+5 z^7 a^{11}+11 z^5 a^{11}-5 z^3 a^{11}-3 z a^{11}+a^{11} z^{-1} -z^{10} a^{10}-14 z^8 a^{10}+30 z^6 a^{10}-20 z^4 a^{10}+3 z^2 a^{10}-4 z^9 a^9-5 z^7 a^9+19 z^5 a^9-18 z^3 a^9+8 z a^9-2 a^9 z^{-1} -6 z^8 a^8+4 z^6 a^8+5 z^4 a^8-8 z^2 a^8+3 a^8-6 z^7 a^7+11 z^5 a^7-12 z^3 a^7+8 z a^7-2 a^7 z^{-1} -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                 7 3     -4 -12               11 5       6 -14             10 8         -2 -16           10 10           0 -18         7 10             3 -20       6 10               -4 -22     2 7                 5 -24   1 6                   -5 -26   2                     2 -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}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-9 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/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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