L11a505

L11a504

L11a506

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Link Presentations

[edit Notes on L11a505's Link Presentations]

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

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{(w-1) \left(u^2 v^2 w^2-2 u^2 v^2 w-u^2 v w^2+3 u^2 v w-u^2 v-u^2 w+u^2-2 u v^2 w^2+3 u v^2 w+2 u v w^2-7 u v w+2 u v+3 u w-2 u+v^2 w^2-v^2 w-v w^2+3 v w-v-2 w+1\right)}{u v w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ -q^8+4 q^7-9 q^6+17 q^5-22 q^4+27 q^3-26 q^2+24 q-17+11 q^{-1} -5 q^{-2} + q^{-3} }[/math] (db)
Signature	2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z^4 a^{-6} -2 z^2 a^{-6} + a^{-6} z^{-2} +2 z^6 a^{-4} +6 z^4 a^{-4} +4 z^2 a^{-4} -2 a^{-4} z^{-2} -2 a^{-4} -z^8 a^{-2} -4 z^6 a^{-2} -5 z^4 a^{-2} -z^2 a^{-2} + a^{-2} z^{-2} +2 a^{-2} +z^6+2 z^4 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ 3 z^{10} a^{-2} +3 z^{10} a^{-4} +9 z^9 a^{-1} +18 z^9 a^{-3} +9 z^9 a^{-5} +18 z^8 a^{-2} +19 z^8 a^{-4} +11 z^8 a^{-6} +10 z^8+5 a z^7-11 z^7 a^{-1} -30 z^7 a^{-3} -6 z^7 a^{-5} +8 z^7 a^{-7} +a^2 z^6-57 z^6 a^{-2} -54 z^6 a^{-4} -16 z^6 a^{-6} +4 z^6 a^{-8} -22 z^6-9 a z^5-8 z^5 a^{-1} +6 z^5 a^{-3} -6 z^5 a^{-5} -10 z^5 a^{-7} +z^5 a^{-9} -a^2 z^4+46 z^4 a^{-2} +48 z^4 a^{-4} +11 z^4 a^{-6} -5 z^4 a^{-8} +13 z^4+3 a z^3+8 z^3 a^{-1} +5 z^3 a^{-3} +5 z^3 a^{-5} +4 z^3 a^{-7} -z^3 a^{-9} -11 z^2 a^{-2} -14 z^2 a^{-4} -3 z^2 a^{-6} +2 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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 17                       1 -1 15                     3   3 13                   6 1   -5 11                 11 3     8 9               12 7       -5 7             15 10         5 5           13 14           1 3         11 13             -2 1       7 14               7 -1     4 10                 -6 -3   1 7                   6 -5   4                     -4 -7 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=-4 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/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=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{13} }[/math] [math]\displaystyle{ {\mathbb Z}^{15} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=7 }[/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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