L11a143

L11a142

L11a144

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Link Presentations

[edit Notes on L11a143's Link Presentations]

Planar diagram presentation	X8192 X18,9,19,10 X6718 X22,19,7,20 X12,5,13,6 X10,4,11,3 X4,15,5,16 X16,12,17,11 X20,13,21,14 X14,21,15,22 X2,18,3,17
Gauss code	{1, -11, 6, -7, 5, -3}, {3, -1, 2, -6, 8, -5, 9, -10, 7, -8, 11, -2, 4, -9, 10, -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 t(1)^2 t(2)^4-3 t(1) t(2)^4+t(2)^4-6 t(1)^2 t(2)^3+11 t(1) t(2)^3-4 t(2)^3+7 t(1)^2 t(2)^2-15 t(1) t(2)^2+7 t(2)^2-4 t(1)^2 t(2)+11 t(1) t(2)-6 t(2)+t(1)^2-3 t(1)+2}{t(1) t(2)^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{27}{q^{9/2}}-\frac{27}{q^{7/2}}+\frac{22}{q^{5/2}}+q^{3/2}-\frac{17}{q^{3/2}}-\frac{1}{q^{19/2}}+\frac{5}{q^{17/2}}-\frac{11}{q^{15/2}}+\frac{18}{q^{13/2}}-\frac{24}{q^{11/2}}-4 \sqrt{q}+\frac{9}{\sqrt{q}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^7 z^5+a^7 z^3-a^7 z-a^7 z^{-1} -a^5 z^7-2 a^5 z^5+a^5 z^3+5 a^5 z+3 a^5 z^{-1} -a^3 z^7-3 a^3 z^5-5 a^3 z^3-6 a^3 z-2 a^3 z^{-1} +a z^5+2 a z^3+a z }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -z^5 a^{11}-5 z^6 a^{10}+4 z^4 a^{10}-11 z^7 a^9+14 z^5 a^9-4 z^3 a^9-14 z^8 a^8+19 z^6 a^8-5 z^4 a^8-2 z^2 a^8+a^8-10 z^9 a^7+3 z^7 a^7+17 z^5 a^7-12 z^3 a^7+3 z a^7-a^7 z^{-1} -3 z^{10} a^6-20 z^8 a^6+47 z^6 a^6-24 z^4 a^6-3 z^2 a^6+3 a^6-17 z^9 a^5+24 z^7 a^5+4 z^5 a^5-17 z^3 a^5+10 z a^5-3 a^5 z^{-1} -3 z^{10} a^4-13 z^8 a^4+37 z^6 a^4-23 z^4 a^4+3 a^4-7 z^9 a^3+6 z^7 a^3+11 z^5 a^3-16 z^3 a^3+9 z a^3-2 a^3 z^{-1} -7 z^8 a^2+13 z^6 a^2-6 z^4 a^2-4 z^7 a+9 z^5 a-7 z^3 a+2 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                 11 3     8 -4               12 7       -5 -6             15 10         5 -8           13 13           0 -10         11 14             -3 -12       7 13               6 -14     4 11                 -7 -16   1 7                   6 -18   4                     -4 -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{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/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}^{11}\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^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{14}\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^{14} }[/math] [math]\displaystyle{ {\mathbb Z}^{15} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/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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