L11a504

L11a503

L11a505

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Link Presentations

[edit Notes on L11a504's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X14,5,15,6 X22,17,13,18 X20,15,21,16 X16,21,17,22 X12,19,7,20 X18,11,19,12 X6718 X4,13,5,14
Gauss code	{1, -2, 3, -11, 4, -10}, {10, -1, 2, -3, 9, -8}, {11, -4, 6, -7, 5, -9, 8, -6, 7, -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{2 u^2 v^2 w^3-2 u^2 v^2 w^2+2 u^2 v^2 w-u^2 v^2-u^2 v w^3+u^2 v w^2-u^2 v w+u^2 v-u v^2 w^3+u v^2 w^2-u v^2 w+u v^2+2 u v w^3-2 u v w^2+2 u v w-2 u v-u w^3+u w^2-u w+u-v w^3+v w^2-v w+v+w^3-2 w^2+2 w-2}{u v w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{-4} - q^{-5} +5 q^{-6} -6 q^{-7} +10 q^{-8} -11 q^{-9} +12 q^{-10} -11 q^{-11} +9 q^{-12} -6 q^{-13} +3 q^{-14} - q^{-15} }[/math] (db)
Signature	-8 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -a^{14} z^{-2} -a^{14}-a^{12} z^6-3 a^{12} z^4+2 a^{12} z^2+4 a^{12} z^{-2} +9 a^{12}+a^{10} z^8+4 a^{10} z^6-a^{10} z^4-18 a^{10} z^2-5 a^{10} z^{-2} -19 a^{10}+a^8 z^8+7 a^8 z^6+18 a^8 z^4+21 a^8 z^2+2 a^8 z^{-2} +11 a^8 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^3 a^{19}+3 z^4 a^{18}+6 z^5 a^{17}-4 z^3 a^{17}+z a^{17}+9 z^6 a^{16}-13 z^4 a^{16}+5 z^2 a^{16}-a^{16}+10 z^7 a^{15}-20 z^5 a^{15}+9 z^3 a^{15}-2 z a^{15}+a^{15} z^{-1} +8 z^8 a^{14}-17 z^6 a^{14}+3 z^4 a^{14}+3 z^2 a^{14}-a^{14} z^{-2} +4 z^9 a^{13}-3 z^7 a^{13}-23 z^5 a^{13}+33 z^3 a^{13}-19 z a^{13}+5 a^{13} z^{-1} +z^{10} a^{12}+6 z^8 a^{12}-32 z^6 a^{12}+37 z^4 a^{12}-20 z^2 a^{12}-4 a^{12} z^{-2} +13 a^{12}+5 z^9 a^{11}-16 z^7 a^{11}-z^5 a^{11}+39 z^3 a^{11}-35 z a^{11}+9 a^{11} z^{-1} +z^{10} a^{10}-z^8 a^{10}-13 z^6 a^{10}+36 z^4 a^{10}-39 z^2 a^{10}-5 a^{10} z^{-2} +22 a^{10}+z^9 a^9-3 z^7 a^9-4 z^5 a^9+20 z^3 a^9-19 z a^9+5 a^9 z^{-1} +z^8 a^8-7 z^6 a^8+18 z^4 a^8-21 z^2 a^8-2 a^8 z^{-2} +11 a^8 }[/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   \ -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -7                       1 1 -9                     1 1 0 -11                   4     4 -13                 2 1     -1 -15               8 4       4 -17             4 3         -1 -19           8 7           1 -21         5 6             1 -23       4 6               -2 -25     2 5                 3 -27   1 4                   -3 -29   2                     2 -31 1                       -1

Integral Khovanov Homology (db, data source)

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