L11n257

L11n256

L11n258

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Link Presentations

[edit Notes on L11n257's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X11,16,12,17 X21,18,22,19 X13,20,14,21 X19,12,20,13 X17,22,18,9 X8,16,5,15 X14,8,15,7 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 9, -8}, {11, -2, -3, 6, -5, -9, 8, 3, -7, 4, -6, 5, -4, 7}

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 v w^2-u v w-3 u w^2+4 u w-2 u+2 v w^3-4 v w^2+3 v w+w^2-w}{\sqrt{u} \sqrt{v} w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 2 q^{-1} -2 q^{-2} +6 q^{-3} -6 q^{-4} +8 q^{-5} -6 q^{-6} +6 q^{-7} -5 q^{-8} +2 q^{-9} - q^{-10} }[/math] (db)
Signature	-2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^{10} z^{-2} -a^{10}+3 z^2 a^8+3 a^8 z^{-2} +5 a^8-2 z^4 a^6-5 z^2 a^6-2 a^6 z^{-2} -5 a^6-z^4 a^4-z^2 a^4-a^4 z^{-2} -2 a^4+2 z^2 a^2+a^2 z^{-2} +3 a^2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{11} z^7-5 a^{11} z^5+9 a^{11} z^3-7 a^{11} z+2 a^{11} z^{-1} +2 a^{10} z^8-8 a^{10} z^6+8 a^{10} z^4-2 a^{10} z^2-a^{10} z^{-2} +a^{10}+a^9 z^9+2 a^9 z^7-24 a^9 z^5+40 a^9 z^3-27 a^9 z+8 a^9 z^{-1} +6 a^8 z^8-23 a^8 z^6+22 a^8 z^4-7 a^8 z^2-3 a^8 z^{-2} +5 a^8+a^7 z^9+5 a^7 z^7-33 a^7 z^5+49 a^7 z^3-34 a^7 z+10 a^7 z^{-1} +4 a^6 z^8-13 a^6 z^6+11 a^6 z^4-4 a^6 z^2-2 a^6 z^{-2} +4 a^6+4 a^5 z^7-13 a^5 z^5+17 a^5 z^3-10 a^5 z+2 a^5 z^{-1} +2 a^4 z^6-3 a^4 z^4+4 a^4 z^2+a^4 z^{-2} -3 a^4+a^3 z^5-a^3 z^3+4 a^3 z-2 a^3 z^{-1} +3 a^2 z^2+a^2 z^{-2} -4 a^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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -1                   2 2 -3                 3 3 0 -5               3   1 4 -7             4 4     0 -9           5 3       2 -11         2 5 1       2 -13       4 4           0 -15     1 2             1 -17   1 4               -3 -19   1                 1 -21 1                   -1

Integral Khovanov Homology (db, data source)

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

L11n256 L11n258