L11a46

L11a45

L11a47

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Link Presentations

[edit Notes on L11a46's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X20,17,21,18 X18,10,19,9 X8,20,9,19 X4,21,1,22 X14,12,15,11 X10,4,11,3 X12,5,13,6 X22,13,5,14 X2,16,3,15
Gauss code	{1, -11, 8, -6}, {9, -1, 2, -5, 4, -8, 7, -9, 10, -7, 11, -2, 3, -4, 5, -3, 6, -10}

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{(t(1)-1) (t(2)-1)^3 \left(t(2)^2-3 t(2)+1\right)}{\sqrt{t(1)} t(2)^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{9/2}-\frac{10}{q^{9/2}}-5 q^{7/2}+\frac{16}{q^{7/2}}+10 q^{5/2}-\frac{22}{q^{5/2}}-17 q^{3/2}+\frac{26}{q^{3/2}}-\frac{1}{q^{13/2}}+\frac{4}{q^{11/2}}+22 \sqrt{q}-\frac{26}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^5 z^3+a^5 z+a^5 z^{-1} -2 a^3 z^5-4 a^3 z^3+z^3 a^{-3} -4 a^3 z-2 a^3 z^{-1} - a^{-3} z^{-1} +a z^7+3 a z^5-2 z^5 a^{-1} +5 a z^3-3 z^3 a^{-1} +3 a z+a z^{-1} + a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^7 z^5-a^7 z^3+4 a^6 z^6-4 a^6 z^4+a^6 z^2+9 a^5 z^7-14 a^5 z^5+11 a^5 z^3-5 a^5 z+a^5 z^{-1} +11 a^4 z^8-15 a^4 z^6+z^6 a^{-4} +9 a^4 z^4-z^4 a^{-4} -3 a^4 z^2+a^4+7 a^3 z^9+7 a^3 z^7+5 z^7 a^{-3} -35 a^3 z^5-10 z^5 a^{-3} +35 a^3 z^3+5 z^3 a^{-3} -15 a^3 z+z a^{-3} +2 a^3 z^{-1} - a^{-3} z^{-1} +2 a^2 z^{10}+22 a^2 z^8+9 z^8 a^{-2} -51 a^2 z^6-19 z^6 a^{-2} +36 a^2 z^4+11 z^4 a^{-2} -12 a^2 z^2-2 z^2 a^{-2} +3 a^2+ a^{-2} +14 a z^9+7 z^9 a^{-1} -9 a z^7-2 z^7 a^{-1} -32 a z^5-22 z^5 a^{-1} +36 a z^3+18 z^3 a^{-1} -12 a z-z a^{-1} +a z^{-1} - a^{-1} z^{-1} +2 z^{10}+20 z^8-52 z^6+35 z^4-10 z^2+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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 10                       1 -1 8                     4   4 6                   6 1   -5 4                 11 4     7 2               11 6       -5 0             15 11         4 -2           13 13           0 -4         9 13             -4 -6       7 13               6 -8     3 9                 -6 -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}^{9}\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^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{13} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{13} }[/math] [math]\displaystyle{ {\mathbb Z}^{15} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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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