L11a110

L11a109

L11a111

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Link Presentations

[edit Notes on L11a110's Link Presentations]

Planar diagram presentation	X6172 X14,4,15,3 X16,8,17,7 X18,10,19,9 X20,12,21,11 X22,14,5,13 X8,18,9,17 X10,20,11,19 X12,22,13,21 X2536 X4,16,1,15
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -7, 4, -8, 5, -9, 6, -2, 11, -3, 7, -4, 8, -5, 9, -6}

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-1) (v-1) \left(v^2+1\right) \left(v^4+1\right)}{\sqrt{u} v^{7/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{25/2}-2 q^{23/2}+3 q^{21/2}-3 q^{19/2}+4 q^{17/2}-4 q^{15/2}+4 q^{13/2}-4 q^{11/2}+2 q^{9/2}-3 q^{7/2}+q^{5/2}-q^{3/2} }[/math] (db)
Signature	7 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^9 a^{-7} +z^7 a^{-5} -8 z^7 a^{-7} +z^7 a^{-9} +7 z^5 a^{-5} -23 z^5 a^{-7} +6 z^5 a^{-9} +16 z^3 a^{-5} -31 z^3 a^{-7} +11 z^3 a^{-9} +14 z a^{-5} -22 z a^{-7} +8 z a^{-9} +4 a^{-5} z^{-1} -7 a^{-7} z^{-1} +3 a^{-9} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^2 a^{-16} +2 z^3 a^{-15} +3 z^4 a^{-14} -3 z^2 a^{-14} + a^{-14} +3 z^5 a^{-13} -4 z^3 a^{-13} +3 z^6 a^{-12} -6 z^4 a^{-12} +3 z^7 a^{-11} -9 z^5 a^{-11} +4 z^3 a^{-11} +3 z^8 a^{-10} -12 z^6 a^{-10} +10 z^4 a^{-10} +3 z^9 a^{-9} -17 z^7 a^{-9} +31 z^5 a^{-9} -25 z^3 a^{-9} +12 z a^{-9} -3 a^{-9} z^{-1} +z^{10} a^{-8} -3 z^8 a^{-8} -7 z^6 a^{-8} +27 z^4 a^{-8} -23 z^2 a^{-8} +7 a^{-8} +4 z^9 a^{-7} -28 z^7 a^{-7} +66 z^5 a^{-7} -65 z^3 a^{-7} +30 z a^{-7} -7 a^{-7} z^{-1} +z^{10} a^{-6} -6 z^8 a^{-6} +8 z^6 a^{-6} +8 z^4 a^{-6} -19 z^2 a^{-6} +7 a^{-6} +z^9 a^{-5} -8 z^7 a^{-5} +23 z^5 a^{-5} -30 z^3 a^{-5} +18 z a^{-5} -4 a^{-5} 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   \ -2 -1 0 1 2 3 4 5 6 7 8 9 χ 26                       1 -1 24                     1   1 22                   2 1   -1 20                 1 1     0 18               3 2       -1 16             1 1         0 14           3 3           0 12         1 1             0 10       1 3               2 8     2 1                 1 6   1 3                   2 4                         0 2 1                       1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=6 }[/math] [math]\displaystyle{ i=8 }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=7 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=8 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=9 }[/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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