L11a315

L11a314

L11a316

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Link Presentations

[edit Notes on L11a315's Link Presentations]

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

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 u^3 v^4-2 u^3 v^3+u^3 v^2+u^2 v^5-4 u^2 v^4+7 u^2 v^3-5 u^2 v^2+2 u^2 v+2 u v^4-5 u v^3+7 u v^2-4 u v+u+v^3-2 v^2+2 v}{u^{3/2} v^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{2}{q^{9/2}}-\frac{1}{q^{7/2}}+\frac{1}{q^{29/2}}-\frac{3}{q^{27/2}}+\frac{6}{q^{25/2}}-\frac{11}{q^{23/2}}+\frac{14}{q^{21/2}}-\frac{15}{q^{19/2}}+\frac{15}{q^{17/2}}-\frac{13}{q^{15/2}}+\frac{9}{q^{13/2}}-\frac{6}{q^{11/2}} }[/math] (db)
Signature	-7 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^3 a^{13}-3 z a^{13}-a^{13} z^{-1} +3 z^5 a^{11}+12 z^3 a^{11}+13 z a^{11}+3 a^{11} z^{-1} -2 z^7 a^9-10 z^5 a^9-16 z^3 a^9-10 z a^9-2 a^9 z^{-1} -z^7 a^7-5 z^5 a^7-8 z^3 a^7-4 z a^7 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{18} z^4-a^{18} z^2+3 a^{17} z^5-3 a^{17} z^3+a^{17} z+5 a^{16} z^6-4 a^{16} z^4+a^{16} z^2+7 a^{15} z^7-9 a^{15} z^5+5 a^{15} z^3+a^{15} z+7 a^{14} z^8-11 a^{14} z^6+8 a^{14} z^4-3 a^{14} z^2+a^{14}+4 a^{13} z^9-13 a^{13} z^5+9 a^{13} z^3+a^{13} z-a^{13} z^{-1} +a^{12} z^{10}+10 a^{12} z^8-35 a^{12} z^6+39 a^{12} z^4-22 a^{12} z^2+3 a^{12}+7 a^{11} z^9-19 a^{11} z^7+20 a^{11} z^5-24 a^{11} z^3+16 a^{11} z-3 a^{11} z^{-1} +a^{10} z^{10}+5 a^{10} z^8-26 a^{10} z^6+32 a^{10} z^4-17 a^{10} z^2+3 a^{10}+3 a^9 z^9-11 a^9 z^7+16 a^9 z^5-17 a^9 z^3+11 a^9 z-2 a^9 z^{-1} +2 a^8 z^8-7 a^8 z^6+6 a^8 z^4+a^7 z^7-5 a^7 z^5+8 a^7 z^3-4 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                   4     4 -12                 5 2     -3 -14               8 4       4 -16             7 5         -2 -18           8 8           0 -20         6 7             1 -22       5 8               -3 -24     2 7                 5 -26   1 4                   -3 -28   2                     2 -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}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-9 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\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^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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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