L11a349

L11a348

L11a350

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Link Presentations

[edit Notes on L11a349's Link Presentations]

Planar diagram presentation	X12,1,13,2 X8493 X14,6,15,5 X18,8,19,7 X20,9,21,10 X10,11,1,12 X6,14,7,13 X4,18,5,17 X22,15,11,16 X2,19,3,20 X16,21,17,22
Gauss code	{1, -10, 2, -8, 3, -7, 4, -2, 5, -6}, {6, -1, 7, -3, 9, -11, 8, -4, 10, -5, 11, -9}

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)-1) (t(2)-1) \left(t(2)^2 t(1)^2-3 t(2) t(1)^2+t(1)^2-3 t(2)^2 t(1)+5 t(2) t(1)-3 t(1)+t(2)^2-3 t(2)+1\right)}{t(1)^{3/2} t(2)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{9/2}-4 q^{7/2}+10 q^{5/2}-18 q^{3/2}+23 \sqrt{q}-\frac{28}{\sqrt{q}}+\frac{27}{q^{3/2}}-\frac{24}{q^{5/2}}+\frac{18}{q^{7/2}}-\frac{10}{q^{9/2}}+\frac{4}{q^{11/2}}-\frac{1}{q^{13/2}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^5 z^3+a^5 z-2 a^3 z^5-3 a^3 z^3+z^3 a^{-3} -a^3 z+z a^{-3} +a z^7+2 a z^5-2 z^5 a^{-1} +a z^3-3 z^3 a^{-1} -z a^{-1} +a z^{-1} - a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -3 a^2 z^{10}-3 z^{10}-10 a^3 z^9-18 a z^9-8 z^9 a^{-1} -13 a^4 z^8-22 a^2 z^8-8 z^8 a^{-2} -17 z^8-9 a^5 z^7+5 a^3 z^7+26 a z^7+8 z^7 a^{-1} -4 z^7 a^{-3} -4 a^6 z^6+21 a^4 z^6+58 a^2 z^6+16 z^6 a^{-2} -z^6 a^{-4} +50 z^6-a^7 z^5+12 a^5 z^5+13 a^3 z^5+8 z^5 a^{-1} +8 z^5 a^{-3} +4 a^6 z^4-17 a^4 z^4-46 a^2 z^4-11 z^4 a^{-2} +2 z^4 a^{-4} -38 z^4+a^7 z^3-8 a^5 z^3-12 a^3 z^3-5 a z^3-8 z^3 a^{-1} -6 z^3 a^{-3} -a^6 z^2+6 a^4 z^2+14 a^2 z^2+3 z^2 a^{-2} -z^2 a^{-4} +11 z^2+2 a^5 z+2 a^3 z+2 z a^{-1} +2 z a^{-3} +1-a z^{-1} - a^{-1} z^{-1} }[/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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 10                       1 -1 8                     3   3 6                   7 1   -6 4                 11 3     8 2               12 7       -5 0             16 11         5 -2           13 14           1 -4         11 14             -3 -6       7 13               6 -8     3 11                 -8 -10   1 7                   6 -12   3                     -3 -14 1                       1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-2 }[/math] [math]\displaystyle{ i=0 }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{13} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{14} }[/math] [math]\displaystyle{ {\mathbb Z}^{16} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/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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