L11n441

L11n440

L11n442

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Link Presentations

[edit Notes on L11n441's Link Presentations]

Planar diagram presentation	X6172 X2536 X18,11,19,12 X10,3,11,4 X4,9,1,10 X14,7,15,8 X8,13,5,14 X19,13,20,22 X15,21,16,20 X21,17,22,16 X12,17,9,18
Gauss code	{1, -2, 4, -5}, {2, -1, 6, -7}, {5, -4, 3, -11}, {7, -6, -9, 10, 11, -3, -8, 9, -10, 8}

A Braid Representative	{{{braid_table}}}
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(1) t(4)^3-t(1) t(2) t(4)^3+t(2) t(4)^3-t(1) t(3) t(4)^3-t(2) t(3) t(4)^3+t(3) t(4)^3-t(4)^3-2 t(1) t(4)^2+t(1) t(2) t(4)^2-t(2) t(4)^2+t(1) t(3) t(4)^2+2 t(2) t(3) t(4)^2-t(3) t(4)^2+2 t(1) t(4)-t(1) t(2) t(4)+t(2) t(4)-t(1) t(3) t(4)-2 t(2) t(3) t(4)+t(3) t(4)-t(1)+t(1) t(2)-t(2)+t(1) t(3)-t(1) t(2) t(3)+2 t(2) t(3)-t(3)}{\sqrt{t(1)} \sqrt{t(2)} \sqrt{t(3)} t(4)^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{12}{q^{9/2}}+\frac{9}{q^{7/2}}-\frac{11}{q^{5/2}}+\frac{9}{q^{3/2}}-\frac{1}{q^{17/2}}+\frac{1}{q^{15/2}}-\frac{6}{q^{13/2}}+\frac{6}{q^{11/2}}+3 \sqrt{q}-\frac{6}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^9 z^{-1} +2 a^9 z^{-3} -4 z a^7-11 a^7 z^{-1} -7 a^7 z^{-3} +6 z^3 a^5+20 z a^5+23 a^5 z^{-1} +9 a^5 z^{-3} -3 z^5 a^3-13 z^3 a^3-22 z a^3-17 a^3 z^{-1} -5 a^3 z^{-3} +3 z^3 a+6 z a+4 a z^{-1} +a z^{-3} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^9 z^7-6 a^9 z^5+14 a^9 z^3-2 a^9 z^{-3} -16 a^9 z+9 a^9 z^{-1} +a^8 z^8-a^8 z^6-10 a^8 z^4+26 a^8 z^2+7 a^8 z^{-2} -23 a^8+a^7 z^9+2 a^7 z^7-21 a^7 z^5+41 a^7 z^3-7 a^7 z^{-3} -39 a^7 z+23 a^7 z^{-1} +6 a^6 z^8-10 a^6 z^6-29 a^6 z^4+73 a^6 z^2+19 a^6 z^{-2} -60 a^6+a^5 z^9+12 a^5 z^7-47 a^5 z^5+51 a^5 z^3-9 a^5 z^{-3} -37 a^5 z+24 a^5 z^{-1} +5 a^4 z^8+a^4 z^6-44 a^4 z^4+75 a^4 z^2+18 a^4 z^{-2} -58 a^4+11 a^3 z^7-29 a^3 z^5+27 a^3 z^3-5 a^3 z^{-3} -16 a^3 z+12 a^3 z^{-1} +10 a^2 z^6-25 a^2 z^4+34 a^2 z^2+7 a^2 z^{-2} -24 a^2+3 a z^5+3 a z^3-a z^{-3} -2 a z+2 a z^{-1} +6 z^2+ z^{-2} -4 }[/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 χ 2                   3 -3 0                 3   3 -2               7 4   -3 -4             4 3 1   2 -6           5 7       2 -8         7 4         3 -10       5 11           6 -12     1 1             0 -14     5               5 -16 1 1                 0 -18 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{ i=0 }[/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} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/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}^{7}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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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