L11a523

L11a522

L11a524

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a523 at Knotilus!

Link Presentations

[edit Notes on L11a523's Link Presentations]

Planar diagram presentation	X8192 X14,4,15,3 X18,7,19,8 X22,9,13,10 X10,21,11,22 X20,14,21,13 X12,17,7,18 X16,6,17,5 X2,11,3,12 X4,16,5,15 X6,20,1,19
Gauss code	{1, -9, 2, -10, 8, -11}, {3, -1, 4, -5, 9, -7}, {6, -2, 10, -8, 7, -3, 11, -6, 5, -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^2 v^2 w^3-2 u^2 v^2 w^2+u^2 v^2 w-2 u^2 v w^3+4 u^2 v w^2-2 u^2 v w-2 u^2 w^2+u^2 w-u v^2 w^3+3 u v^2 w^2-3 u v^2 w+u v^2+2 u v w^3-7 u v w^2+7 u v w-2 u v-u w^3+3 u w^2-3 u w+u-v^2 w^2+2 v^2 w+2 v w^2-4 v w+2 v-w^2+2 w-1}{u v w^{3/2}} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{-6} -q^5-3 q^{-5} +4 q^4+8 q^{-4} -8 q^3-12 q^{-3} +14 q^2+18 q^{-2} -18 q-20 q^{-1} +21 }[/math] (db)
Signature	0 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^4 z^4+3 a^4 z^2+a^4 z^{-2} +3 a^4-2 a^2 z^6-z^6 a^{-2} -8 a^2 z^4-3 z^4 a^{-2} -11 a^2 z^2-2 z^2 a^{-2} -2 a^2 z^{-2} -6 a^2+ a^{-2} +z^8+5 z^6+9 z^4+6 z^2+ z^{-2} +2 }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^6 z^6-3 a^6 z^4+2 a^6 z^2+3 a^5 z^7-7 a^5 z^5+z^5 a^{-5} +3 a^5 z^3-z^3 a^{-5} +6 a^4 z^8-18 a^4 z^6+4 z^6 a^{-4} +23 a^4 z^4-6 z^4 a^{-4} -19 a^4 z^2+2 z^2 a^{-4} -a^4 z^{-2} +7 a^4+5 a^3 z^9-7 a^3 z^7+7 z^7 a^{-3} -5 a^3 z^5-10 z^5 a^{-3} +13 a^3 z^3+3 z^3 a^{-3} -9 a^3 z+2 a^3 z^{-1} +2 a^2 z^{10}+8 a^2 z^8+8 z^8 a^{-2} -31 a^2 z^6-10 z^6 a^{-2} +39 a^2 z^4+z^4 a^{-2} -27 a^2 z^2+3 z^2 a^{-2} -2 a^2 z^{-2} +11 a^2-2 a^{-2} +11 a z^9+6 z^9 a^{-1} -21 a z^7-4 z^7 a^{-1} +10 a z^5-3 z^5 a^{-1} +8 a z^3+2 z^3 a^{-1} -9 a z+2 a z^{-1} +2 z^{10}+10 z^8-26 z^6+20 z^4-5 z^2- z^{-2} +3 }[/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                     3   3 7                   5 1   -4 5                 9 3     6 3               10 6       -4 1             11 8         3 -1           11 12           1 -3         7 9             -2 -5       5 11               6 -7     3 7                 -4 -9     5                   5 -11 1 3                     -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{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\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^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\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^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/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.

L11a522 L11a524