L11a497

L11a496

L11a498

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Link Presentations

[edit Notes on L11a497's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X14,8,15,7 X20,11,21,12 X18,10,5,9 X10,19,11,20 X8,16,9,15 X16,21,17,22 X22,17,19,18 X2536 X4,14,1,13
Gauss code	{1, -10, 2, -11}, {6, -4, 8, -9}, {10, -1, 3, -7, 5, -6, 4, -2, 11, -3, 7, -8, 9, -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{(u-1) (w-1) \left(v w^4-2 v w^3+2 v w^2-2 v w+2 w^3-2 w^2+2 w-1\right)}{\sqrt{u} \sqrt{v} w^{5/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{-6} -q^5-3 q^{-5} +3 q^4+8 q^{-4} -7 q^3-11 q^{-3} +11 q^2+16 q^{-2} -15 q-17 q^{-1} +19 }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^4 z^4+3 a^4 z^2+2 a^4 z^{-2} +4 a^4-2 a^2 z^6-z^6 a^{-2} -9 a^2 z^4-4 z^4 a^{-2} -16 a^2 z^2-6 z^2 a^{-2} -5 a^2 z^{-2} - a^{-2} z^{-2} -14 a^2-4 a^{-2} +z^8+6 z^6+15 z^4+19 z^2+4 z^{-2} +14 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^2 z^{10}+z^{10}+4 a^3 z^9+8 a z^9+4 z^9 a^{-1} +5 a^4 z^8+13 a^2 z^8+6 z^8 a^{-2} +14 z^8+3 a^5 z^7-3 a^3 z^7-10 a z^7+z^7 a^{-1} +5 z^7 a^{-3} +a^6 z^6-12 a^4 z^6-49 a^2 z^6-11 z^6 a^{-2} +3 z^6 a^{-4} -50 z^6-7 a^5 z^5-14 a^3 z^5-19 a z^5-21 z^5 a^{-1} -8 z^5 a^{-3} +z^5 a^{-5} -3 a^6 z^4+10 a^4 z^4+68 a^2 z^4+12 z^4 a^{-2} -5 z^4 a^{-4} +72 z^4+3 a^5 z^3+22 a^3 z^3+50 a z^3+37 z^3 a^{-1} +4 z^3 a^{-3} -2 z^3 a^{-5} +3 a^6 z^2-13 a^4 z^2-50 a^2 z^2-11 z^2 a^{-2} +z^2 a^{-4} -46 z^2-17 a^3 z-35 a z-23 z a^{-1} -4 z a^{-3} +z a^{-5} -a^6+8 a^4+22 a^2+5 a^{-2} +19+5 a^3 z^{-1} +9 a z^{-1} +5 a^{-1} z^{-1} + a^{-3} z^{-1} -2 a^4 z^{-2} -5 a^2 z^{-2} - a^{-2} z^{-2} -4 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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 11                       1 -1 9                     2   2 7                   5 1   -4 5                 6 2     4 3               9 5       -4 1             10 6         4 -1           9 11           2 -3         7 8             -1 -5       4 9               5 -7     4 7                 -3 -9   1 6                   5 -11   2                     -2 -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}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/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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