L10a174

L10a173

L10n1

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a174 at Knotilus!
	L10a174 is a closed five-link chain.

Five linked circles.		Decorative pentagonal representation.		(alternate version)		Highly ornamental version.
Quasi-yin-yang.

Link Presentations

[edit Notes on L10a174'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 X20,15,17,16 X16,19,13,20 X12,17,9,18
Gauss code	{1, -2, 4, -5}, {2, -1, 6, -7}, {5, -4, 3, -10}, {7, -6, 8, -9}, {10, -3, 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{u v w x+u v w y-u v w+u v x y-u v x-2 u v y+u v+u w x y-2 u w x-u w y+u w-2 u x y+2 u x+2 u y-u+v w x y-2 v w x-2 v w y+2 v w-v x y+v x+2 v y-v-w x y+2 w x+w y-w+x y-x-y}{\sqrt{u} \sqrt{v} \sqrt{w} \sqrt{x} \sqrt{y}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{-12} - q^{-11} +6 q^{-10} -6 q^{-9} +15 q^{-8} -11 q^{-7} +15 q^{-6} -10 q^{-5} +10 q^{-4} -4 q^{-3} + q^{-2} }[/math] (db)
Signature	-4 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^{14} z^{-4} -4 a^{12} z^{-4} -5 a^{12} z^{-2} +6 a^{10} z^{-4} +15 a^{10} z^{-2} +10 a^{10}-4 a^8 z^{-4} -10 a^8 z^2-15 a^8 z^{-2} -20 a^8+4 a^6 z^4+a^6 z^{-4} +10 a^6 z^2+5 a^6 z^{-2} +10 a^6+a^4 z^4 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{14} z^6-5 a^{14} z^4-a^{14} z^{-4} +10 a^{14} z^2+5 a^{14} z^{-2} -10 a^{14}+a^{13} z^7-10 a^{13} z^3+4 a^{13} z^{-3} +20 a^{13} z-15 a^{13} z^{-1} +a^{12} z^8+4 a^{12} z^6-20 a^{12} z^4-4 a^{12} z^{-4} +30 a^{12} z^2+14 a^{12} z^{-2} -25 a^{12}+a^{11} z^9+2 a^{11} z^7+2 a^{11} z^5-30 a^{11} z^3+12 a^{11} z^{-3} +55 a^{11} z-41 a^{11} z^{-1} +6 a^{10} z^8-2 a^{10} z^6-25 a^{10} z^4-6 a^{10} z^{-4} +40 a^{10} z^2+18 a^{10} z^{-2} -31 a^{10}+a^9 z^9+11 a^9 z^7-12 a^9 z^5-30 a^9 z^3+12 a^9 z^{-3} +55 a^9 z-41 a^9 z^{-1} +5 a^8 z^8+5 a^8 z^6-25 a^8 z^4-4 a^8 z^{-4} +30 a^8 z^2+14 a^8 z^{-2} -25 a^8+10 a^7 z^7-10 a^7 z^5-10 a^7 z^3+4 a^7 z^{-3} +20 a^7 z-15 a^7 z^{-1} +10 a^6 z^6-14 a^6 z^4-a^6 z^{-4} +10 a^6 z^2+5 a^6 z^{-2} -10 a^6+4 a^5 z^5+a^4 z^4 }[/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   \ -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -3                     1 1 -5                   4 1 -3 -7                 6     6 -9               4 4     0 -11             11 6       5 -13           10 14         4 -15         5 1           4 -17       1 10             9 -19     5 5               0 -21   1 6                 5 -23                       0 -25 1                     1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-5 }[/math] [math]\displaystyle{ i=-3 }[/math] [math]\displaystyle{ r=-10 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-9 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{14}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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.

L10a173 L10n1