L10a56

L10a55

L10a57

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Link Presentations

[edit Notes on L10a56's Link Presentations]

Planar diagram presentation	X8192 X16,7,17,8 X10,4,11,3 X12,5,13,6 X20,12,7,11 X18,13,19,14 X2,15,3,16 X4,19,5,20 X14,10,15,9 X6,18,1,17
Gauss code	{1, -7, 3, -8, 4, -10}, {2, -1, 9, -3, 5, -4, 6, -9, 7, -2, 10, -6, 8, -5}

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^2 v^4-4 u^2 v^3+6 u^2 v^2-4 u^2 v+u^2-2 u v^4+8 u v^3-13 u v^2+8 u v-2 u+v^4-4 v^3+6 v^2-4 v+1}{u v^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -\frac{11}{q^{9/2}}-q^{7/2}+\frac{16}{q^{7/2}}+5 q^{5/2}-\frac{21}{q^{5/2}}-11 q^{3/2}+\frac{22}{q^{3/2}}-\frac{1}{q^{13/2}}+\frac{5}{q^{11/2}}+16 \sqrt{q}-\frac{21}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a z^7-2 a^3 z^5+3 a z^5-z^5 a^{-1} +a^5 z^3-3 a^3 z^3+3 a z^3-z^3 a^{-1} -a z+a^3 z^{-1} -a z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ -5 a^3 z^9-5 a z^9-12 a^4 z^8-24 a^2 z^8-12 z^8-11 a^5 z^7-14 a^3 z^7-14 a z^7-11 z^7 a^{-1} -5 a^6 z^6+16 a^4 z^6+42 a^2 z^6-5 z^6 a^{-2} +16 z^6-a^7 z^5+16 a^5 z^5+39 a^3 z^5+39 a z^5+16 z^5 a^{-1} -z^5 a^{-3} +4 a^6 z^4-4 a^4 z^4-16 a^2 z^4+4 z^4 a^{-2} -4 z^4-6 a^5 z^3-18 a^3 z^3-18 a z^3-6 z^3 a^{-1} -a^3 z-a z-a^2+a^3 z^{-1} +a 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 χ 8                     1 1 6                   4   -4 4                 7 1   6 2               9 4     -5 0             12 7       5 -2           11 10         -1 -4         10 11           -1 -6       7 12             5 -8     4 9               -5 -10   1 7                 6 -12   4                   -4 -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}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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}^{12}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/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=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=4 }[/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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