L11a170

L11a169

L11a171

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a170 at Knotilus!

Link Presentations

[edit Notes on L11a170's Link Presentations]

Planar diagram presentation	X8192 X10,4,11,3 X12,7,13,8 X22,15,7,16 X14,6,15,5 X6,14,1,13 X16,21,17,22 X18,10,19,9 X20,11,21,12 X4,18,5,17 X2,19,3,20
Gauss code	{1, -11, 2, -10, 5, -6}, {3, -1, 8, -2, 9, -3, 6, -5, 4, -7, 10, -8, 11, -9, 7, -4}

A Braid Representative	{{{braid_table}}}
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^4-5 u^2 v^3+8 u^2 v^2-6 u^2 v+2 u^2-3 u v^4+11 u v^3-15 u v^2+11 u v-3 u+2 v^4-6 v^3+8 v^2-5 v+1}{u v^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ q^{9/2}-\frac{10}{q^{9/2}}-5 q^{7/2}+\frac{18}{q^{7/2}}+11 q^{5/2}-\frac{24}{q^{5/2}}-19 q^{3/2}+\frac{28}{q^{3/2}}-\frac{1}{q^{13/2}}+\frac{4}{q^{11/2}}+24 \sqrt{q}-\frac{29}{\sqrt{q}} }[/math] (db)
Signature	-1 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a z^7-2 a^3 z^5+2 a z^5-2 z^5 a^{-1} +a^5 z^3-3 a^3 z^3+a z^3-2 z^3 a^{-1} +z^3 a^{-3} +a^5 z-a^3 z-a z+2 z a^{-1} + a^{-1} z^{-1} - a^{-3} z^{-1} }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^7 z^5-a^7 z^3+4 a^6 z^6-4 a^6 z^4+a^6 z^2+9 a^5 z^7-12 a^5 z^5+7 a^5 z^3-a^5 z+13 a^4 z^8-21 a^4 z^6+z^6 a^{-4} +17 a^4 z^4-z^4 a^{-4} -6 a^4 z^2+11 a^3 z^9-11 a^3 z^7+5 z^7 a^{-3} -9 z^5 a^{-3} +2 a^3 z^3+4 z^3 a^{-3} +a^3 z+z a^{-3} - a^{-3} z^{-1} +4 a^2 z^{10}+17 a^2 z^8+10 z^8 a^{-2} -48 a^2 z^6-20 z^6 a^{-2} +38 a^2 z^4+10 z^4 a^{-2} -12 a^2 z^2-z^2 a^{-2} + a^{-2} +21 a z^9+10 z^9 a^{-1} -39 a z^7-14 z^7 a^{-1} +21 a z^5-z^5 a^{-1} -10 a z^3+5 a z+4 z a^{-1} - a^{-1} z^{-1} +4 z^{10}+14 z^8-44 z^6+28 z^4-6 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 10                       1 -1 8                     4   4 6                   7 1   -6 4                 12 4     8 2               13 8       -5 0             16 11         5 -2           13 14           1 -4         11 15             -4 -6       7 13               6 -8     3 11                 -8 -10   1 7                   6 -12   3                     -3 -14 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=0 }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{15}\oplus{\mathbb Z}_2^{13} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{15} }[/math] [math]\displaystyle{ {\mathbb Z}^{16} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{13} }[/math] [math]\displaystyle{ {\mathbb Z}^{13} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{12} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/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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