L11a485

L11a484

L11a486

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Link Presentations

[edit Notes on L11a485's Link Presentations]

Planar diagram presentation	X6172 X14,7,15,8 X4,15,1,16 X12,6,13,5 X8493 X22,11,19,12 X18,22,5,21 X20,10,21,9 X10,17,11,18 X16,19,17,20 X2,14,3,13
Gauss code	{1, -11, 5, -3}, {10, -8, 7, -6}, {4, -1, 2, -5, 8, -9, 6, -4, 11, -2, 3, -10, 9, -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-1) (v-1) (w-1)^3 \left(w^2-w+1\right)}{\sqrt{u} \sqrt{v} w^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{-6} -q^5-6 q^{-5} +5 q^4+13 q^{-4} -11 q^3-20 q^{-3} +19 q^2+28 q^{-2} -26 q-30 q^{-1} +32 }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^4 z^4-2 a^4-2 a^2 z^6-z^6 a^{-2} -4 a^2 z^4-2 z^4 a^{-2} +2 a^2 z^2+a^2 z^{-2} + a^{-2} z^{-2} +6 a^2+2 a^{-2} +z^8+4 z^6+5 z^4-2 z^2-2 z^{-2} -6 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^6 z^6+6 a^5 z^7-8 a^5 z^5+z^5 a^{-5} +2 a^5 z+13 a^4 z^8-26 a^4 z^6+5 z^6 a^{-4} +12 a^4 z^4-4 z^4 a^{-4} +2 a^4 z^2-4 a^4+12 a^3 z^9-14 a^3 z^7+11 z^7 a^{-3} -8 a^3 z^5-13 z^5 a^{-3} +2 a^3 z^3+4 z^3 a^{-3} +6 a^3 z+4 a^2 z^{10}+24 a^2 z^8+15 z^8 a^{-2} -69 a^2 z^6-20 z^6 a^{-2} +38 a^2 z^4+8 z^4 a^{-2} +10 a^2 z^2+2 z^2 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} -12 a^2-4 a^{-2} +24 a z^9+12 z^9 a^{-1} -36 a z^7-5 z^7 a^{-1} -3 a z^5-17 z^5 a^{-1} +10 a z^3+12 z^3 a^{-1} +6 a z+2 z a^{-1} -2 a z^{-1} -2 a^{-1} z^{-1} +4 z^{10}+26 z^8-67 z^6+38 z^4+10 z^2+2 z^{-2} -11 }[/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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 11                       1 -1 9                     4   4 7                   7 1   -6 5                 12 4     8 3               14 7       -7 1             18 12         6 -1           16 18           2 -3         12 14             -2 -5       8 16               8 -7     5 12                 -7 -9   1 8                   7 -11   5                     -5 -13 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=1 }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{16}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{16} }[/math] [math]\displaystyle{ {\mathbb Z}^{16} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{18}\oplus{\mathbb Z}_2^{14} }[/math] [math]\displaystyle{ {\mathbb Z}^{18} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{14} }[/math] [math]\displaystyle{ {\mathbb Z}^{14} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{12} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=5 }[/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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