L11a107

L11a106

L11a108

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Link Presentations

[edit Notes on L11a107's Link Presentations]

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

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(2)^7-2 t(1) t(2)^6+2 t(2)^6+2 t(1) t(2)^5-2 t(2)^5-2 t(1) t(2)^4+2 t(2)^4+2 t(1) t(2)^3-2 t(2)^3-2 t(1) t(2)^2+2 t(2)^2+2 t(1) t(2)-2 t(2)-t(1)}{\sqrt{t(1)} t(2)^{7/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{2}{q^{9/2}}-\frac{1}{q^{7/2}}+\frac{1}{q^{29/2}}-\frac{2}{q^{27/2}}+\frac{4}{q^{25/2}}-\frac{5}{q^{23/2}}+\frac{7}{q^{21/2}}-\frac{8}{q^{19/2}}+\frac{7}{q^{17/2}}-\frac{7}{q^{15/2}}+\frac{4}{q^{13/2}}-\frac{4}{q^{11/2}} }[/math] (db)
Signature	-7 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^3 a^{13}-4 z a^{13}-3 a^{13} z^{-1} +3 z^5 a^{11}+15 z^3 a^{11}+20 z a^{11}+7 a^{11} z^{-1} -2 z^7 a^9-12 z^5 a^9-23 z^3 a^9-17 z a^9-4 a^9 z^{-1} -z^7 a^7-5 z^5 a^7-6 z^3 a^7-z a^7 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{18} z^4-2 a^{18} z^2+a^{18}+2 a^{17} z^5-3 a^{17} z^3+2 a^{16} z^6-a^{16} z^4-2 a^{16} z^2+2 a^{15} z^7-2 a^{15} z^5+a^{15} z^3+2 a^{14} z^8-4 a^{14} z^6+4 a^{14} z^4+2 a^{13} z^9-7 a^{13} z^7+12 a^{13} z^5-12 a^{13} z^3+10 a^{13} z-3 a^{13} z^{-1} +a^{12} z^{10}-a^{12} z^8-9 a^{12} z^6+24 a^{12} z^4-22 a^{12} z^2+7 a^{12}+5 a^{11} z^9-27 a^{11} z^7+55 a^{11} z^5-57 a^{11} z^3+29 a^{11} z-7 a^{11} z^{-1} +a^{10} z^{10}-a^{10} z^8-12 a^{10} z^6+27 a^{10} z^4-23 a^{10} z^2+7 a^{10}+3 a^9 z^9-17 a^9 z^7+34 a^9 z^5-35 a^9 z^3+18 a^9 z-4 a^9 z^{-1} +2 a^8 z^8-9 a^8 z^6+9 a^8 z^4-a^8 z^2+a^7 z^7-5 a^7 z^5+6 a^7 z^3-a^7 z }[/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 -6                       1 1 -8                     2 1 -1 -10                   2     2 -12                 2 2     0 -14               5 2       3 -16             3 3         0 -18           5 4           1 -20         2 3             1 -22       3 5               -2 -24     1 2                 1 -26   1 3                   -2 -28   1                     1 -30 1                       -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-8 }[/math] [math]\displaystyle{ i=-6 }[/math] [math]\displaystyle{ r=-11 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-10 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-9 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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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