L11a507

L11a506

L11a508

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Link Presentations

[edit Notes on L11a507's Link Presentations]

Planar diagram presentation	X8192 X18,12,19,11 X10,4,11,3 X2,20,3,19 X16,8,17,7 X20,9,21,10 X12,18,7,17 X22,16,13,15 X14,6,15,5 X4,14,5,13 X6,21,1,22
Gauss code	{1, -4, 3, -10, 9, -11}, {5, -1, 6, -3, 2, -7}, {10, -9, 8, -5, 7, -2, 4, -6, 11, -8}

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{(w-1) \left(u^2 v^2 w+u^2 \left(-v^2\right)+u^2 v w^2-2 u^2 v w+u^2 v-u^2 w^2+u^2 w-2 u v^2 w+2 u v^2-2 u v w^2+5 u v w-2 u v+2 u w^2-2 u w+v^2 w-v^2+v w^2-2 v w+v-w^2+w\right)}{u v w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^9-4 q^8+8 q^7-13 q^6+19 q^5-20 q^4+22 q^3-18 q^2- q^{-2} +14 q+4 q^{-1} -8 }[/math] (db)
Signature	2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ z^2 a^{-8} -2 z^4 a^{-6} -2 z^2 a^{-6} + a^{-6} z^{-2} +z^6 a^{-4} +z^4 a^{-4} -2 a^{-4} z^{-2} -2 a^{-4} +z^6 a^{-2} +2 z^4 a^{-2} +3 z^2 a^{-2} + a^{-2} z^{-2} +2 a^{-2} -z^4-z^2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ 2 z^{10} a^{-4} +2 z^{10} a^{-6} +6 z^9 a^{-3} +12 z^9 a^{-5} +6 z^9 a^{-7} +8 z^8 a^{-2} +12 z^8 a^{-4} +11 z^8 a^{-6} +7 z^8 a^{-8} +7 z^7 a^{-1} -3 z^7 a^{-3} -24 z^7 a^{-5} -10 z^7 a^{-7} +4 z^7 a^{-9} -10 z^6 a^{-2} -35 z^6 a^{-4} -41 z^6 a^{-6} -19 z^6 a^{-8} +z^6 a^{-10} +4 z^6+a z^5-10 z^5 a^{-1} -6 z^5 a^{-3} +14 z^5 a^{-5} -z^5 a^{-7} -10 z^5 a^{-9} +2 z^4 a^{-2} +34 z^4 a^{-4} +44 z^4 a^{-6} +16 z^4 a^{-8} -2 z^4 a^{-10} -6 z^4-a z^3+3 z^3 a^{-1} +4 z^3 a^{-3} +5 z^3 a^{-7} +5 z^3 a^{-9} +z^2 a^{-2} -10 z^2 a^{-4} -15 z^2 a^{-6} -6 z^2 a^{-8} +2 z^2+2 z a^{-3} +2 z a^{-5} -2 a^{-2} -3 a^{-4} -2 a^{-6} -2 a^{-3} z^{-1} -2 a^{-5} z^{-1} + a^{-2} z^{-2} +2 a^{-4} z^{-2} + a^{-6} 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   \ -3 -2 -1 0 1 2 3 4 5 6 7 8 χ 19                       1 1 17                     3   -3 15                   5 1   4 13                 8 3     -5 11               11 5       6 9             10 9         -1 7           12 10           2 5         8 12             4 3       6 10               -4 1     3 9                 6 -1   1 5                   -4 -3   3                     3 -5 1                       -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=1 }[/math] [math]\displaystyle{ i=3 }[/math] [math]\displaystyle{ r=-3 }[/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=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=7 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=8 }[/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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