L11a335

L11a334

L11a336

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Link Presentations

[edit Notes on L11a335's Link Presentations]

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

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{u^3 v^4-3 u^3 v^3+3 u^3 v^2+u^2 v^5-5 u^2 v^4+12 u^2 v^3-11 u^2 v^2+4 u^2 v+4 u v^4-11 u v^3+12 u v^2-5 u v+u+3 v^3-3 v^2+v}{u^{3/2} v^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{26}{q^{9/2}}-\frac{26}{q^{7/2}}+\frac{22}{q^{5/2}}+q^{3/2}-\frac{16}{q^{3/2}}-\frac{1}{q^{19/2}}+\frac{4}{q^{17/2}}-\frac{11}{q^{15/2}}+\frac{17}{q^{13/2}}-\frac{23}{q^{11/2}}-4 \sqrt{q}+\frac{9}{\sqrt{q}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^5 a^7+2 z^3 a^7+3 z a^7+2 a^7 z^{-1} -z^7 a^5-3 z^5 a^5-5 z^3 a^5-6 z a^5-3 a^5 z^{-1} -z^7 a^3-3 z^5 a^3-4 z^3 a^3-2 z a^3+a^3 z^{-1} +z^5 a+2 z^3 a+z a }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^5 a^{11}+z^3 a^{11}-4 z^6 a^{10}+3 z^4 a^{10}-10 z^7 a^9+16 z^5 a^9-13 z^3 a^9+6 z a^9-13 z^8 a^8+20 z^6 a^8-10 z^4 a^8+z^2 a^8-10 z^9 a^7+9 z^7 a^7+z^5 a^7+3 z^3 a^7-6 z a^7+2 a^7 z^{-1} -3 z^{10} a^6-18 z^8 a^6+48 z^6 a^6-39 z^4 a^6+15 z^2 a^6-3 a^6-17 z^9 a^5+31 z^7 a^5-20 z^5 a^5+16 z^3 a^5-12 z a^5+3 a^5 z^{-1} -3 z^{10} a^4-12 z^8 a^4+39 z^6 a^4-36 z^4 a^4+17 z^2 a^4-3 a^4-7 z^9 a^3+8 z^7 a^3+5 z^5 a^3-7 z^3 a^3+z a^3+a^3 z^{-1} -7 z^8 a^2+14 z^6 a^2-8 z^4 a^2+2 z^2 a^2-a^2-4 z^7 a+9 z^5 a-6 z^3 a+z a-z^6+2 z^4-z^2 }[/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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 4                       1 -1 2                     3   3 0                   6 1   -5 -2                 10 3     7 -4               13 7       -6 -6             13 9         4 -8           13 13           0 -10         10 13             -3 -12       7 13               6 -14     4 10                 -6 -16     7                   7 -18 1 4                     -3 -20 1                       1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-4 }[/math] [math]\displaystyle{ i=-2 }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{13} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{13} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{13} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=3 }[/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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