L10a17

L10a16

L10a18

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Link Presentations

[edit Notes on L10a17's Link Presentations]

Planar diagram presentation	X6172 X10,4,11,3 X16,8,17,7 X20,18,5,17 X18,14,19,13 X14,20,15,19 X12,10,13,9 X8,16,9,15 X2536 X4,12,1,11
Gauss code	{1, -9, 2, -10}, {9, -1, 3, -8, 7, -2, 10, -7, 5, -6, 8, -3, 4, -5, 6, -4}

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) \left(3 v^2-4 v+3\right)}{\sqrt{u} v^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ 13 q^{9/2}-13 q^{7/2}+9 q^{5/2}-6 q^{3/2}-q^{19/2}+4 q^{17/2}-7 q^{15/2}+11 q^{13/2}-13 q^{11/2}+2 \sqrt{q}-\frac{1}{\sqrt{q}} }[/math] (db)
Signature	3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ -z a^{-9} +3 z^3 a^{-7} +4 z a^{-7} + a^{-7} z^{-1} -2 z^5 a^{-5} -5 z^3 a^{-5} -5 z a^{-5} -2 a^{-5} z^{-1} -z^5 a^{-3} -z^3 a^{-3} +z^3 a^{-1} +2 z a^{-1} + a^{-1} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ z^5 a^{-11} -z^3 a^{-11} +4 z^6 a^{-10} -7 z^4 a^{-10} +3 z^2 a^{-10} +6 z^7 a^{-9} -10 z^5 a^{-9} +4 z^3 a^{-9} -2 z a^{-9} +4 z^8 a^{-8} +2 z^6 a^{-8} -17 z^4 a^{-8} +12 z^2 a^{-8} -2 a^{-8} +z^9 a^{-7} +12 z^7 a^{-7} -29 z^5 a^{-7} +23 z^3 a^{-7} -8 z a^{-7} + a^{-7} z^{-1} +7 z^8 a^{-6} -5 z^6 a^{-6} -10 z^4 a^{-6} +14 z^2 a^{-6} -5 a^{-6} +z^9 a^{-5} +9 z^7 a^{-5} -22 z^5 a^{-5} +21 z^3 a^{-5} -9 z a^{-5} +2 a^{-5} z^{-1} +3 z^8 a^{-4} -z^6 a^{-4} -3 z^4 a^{-4} +5 z^2 a^{-4} -3 a^{-4} +3 z^7 a^{-3} -3 z^5 a^{-3} +2 z^6 a^{-2} -3 z^4 a^{-2} + a^{-2} +z^5 a^{-1} -3 z^3 a^{-1} +3 z a^{-1} - a^{-1} z^{-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   \ -2 -1 0 1 2 3 4 5 6 7 8 χ 20                     1 1 18                   3   -3 16                 4 1   3 14               7 3     -4 12             6 4       2 10           7 7         0 8         6 6           0 6       3 7             4 4     3 6               -3 2   1 5                 4 0   1                   -1 -2 1                     1

Integral Khovanov Homology (db, data source)

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

L10a16 L10a18