L11a548

L11a547

L11n1

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Link L11a548.

A graph, L11a548.

Link Presentations

[edit Notes on L11a548's Link Presentations]

Planar diagram presentation	X6172 X2536 X18,11,19,12 X10,3,11,4 X4,9,1,10 X14,7,15,8 X8,13,5,14 X22,20,17,19 X16,22,13,21 X20,16,21,15 X12,17,9,18
Gauss code	{1, -2, 4, -5}, {2, -1, 6, -7}, {5, -4, 3, -11}, {7, -6, 10, -9}, {11, -3, 8, -10, 9, -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{-2 t(2) t(1)+2 t(2) t(3) t(1)-2 t(3) t(1)+t(2) t(4) t(1)-t(2) t(3) t(4) t(1)+2 t(3) t(4) t(1)-2 t(4) t(1)+2 t(2) t(5) t(1)-t(2) t(3) t(5) t(1)+t(3) t(5) t(1)-2 t(2) t(4) t(5) t(1)+t(2) t(3) t(4) t(5) t(1)-2 t(3) t(4) t(5) t(1)+3 t(4) t(5) t(1)-2 t(5) t(1)+2 t(1)+2 t(2)-3 t(2) t(3)+2 t(3)-t(2) t(4)+2 t(2) t(3) t(4)-2 t(3) t(4)+t(4)-2 t(2) t(5)+2 t(2) t(3) t(5)-t(3) t(5)+2 t(2) t(4) t(5)-2 t(2) t(3) t(4) t(5)+2 t(3) t(4) t(5)-2 t(4) t(5)+t(5)-1}{\sqrt{t(1)} \sqrt{t(2)} \sqrt{t(3)} \sqrt{t(4)} \sqrt{t(5)}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{-9} - q^{-8} +6 q^{-7} -6 q^{-6} +16 q^{-5} -15 q^{-4} +21 q^{-3} -q^2-15 q^{-2} +5 q+15 q^{-1} -10 }[/math] (db)
Signature	-2 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^{10} z^{-2} +a^{10} z^{-4} -7 a^8 z^{-2} -4 a^8 z^{-4} -4 a^8+6 z^2 a^6+15 a^6 z^{-2} +6 a^6 z^{-4} +14 a^6-4 z^4 a^4-10 z^2 a^4-13 a^4 z^{-2} -4 a^4 z^{-4} -16 a^4+z^6 a^2+2 z^4 a^2+4 z^2 a^2+4 a^2 z^{-2} +a^2 z^{-4} +6 a^2-z^4 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^6 a^{10}-5 z^4 a^{10}+10 z^2 a^{10}+5 a^{10} z^{-2} -a^{10} z^{-4} -10 a^{10}+z^7 a^9-10 z^3 a^9+20 z a^9-15 a^9 z^{-1} +4 a^9 z^{-3} +z^8 a^8+4 z^6 a^8-20 z^4 a^8+30 z^2 a^8+14 a^8 z^{-2} -4 a^8 z^{-4} -25 a^8+z^9 a^7+3 z^7 a^7-30 z^3 a^7+55 z a^7-41 a^7 z^{-1} +12 a^7 z^{-3} +z^{10} a^6+z^8 a^6+9 z^6 a^6-31 z^4 a^6+40 z^2 a^6+18 a^6 z^{-2} -6 a^6 z^{-4} -31 a^6+6 z^9 a^5-8 z^7 a^5+11 z^5 a^5-30 z^3 a^5+55 z a^5-41 a^5 z^{-1} +12 a^5 z^{-3} +z^{10} a^4+10 z^8 a^4-14 z^6 a^4-11 z^4 a^4+30 z^2 a^4+14 a^4 z^{-2} -4 a^4 z^{-4} -25 a^4+5 z^9 a^3-5 z^5 a^3-10 z^3 a^3+20 z a^3-15 a^3 z^{-1} +4 a^3 z^{-3} +10 z^8 a^2-15 z^6 a^2+10 z^2 a^2+5 a^2 z^{-2} -a^2 z^{-4} -10 a^2+10 z^7 a-15 z^5 a+5 z^6-5 z^4+z^5 a^{-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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 5                       1 -1 3                     4   4 1                   6 1   -5 -1                 9 4     5 -3               11 11       0 -5             10 4         6 -7           5 11           6 -9         11 10             1 -11       5 15               10 -13     1 1                 0 -15     5                   5 -17 1 1                     0 -19 1                       1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-3 }[/math] [math]\displaystyle{ i=-1 }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{15}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/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=3 }[/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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