L11a357

L11a356

L11a358

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a357 at Knotilus!

Link Presentations

[edit Notes on L11a357's Link Presentations]

Planar diagram presentation	X12,1,13,2 X18,7,19,8 X14,3,15,4 X20,10,21,9 X22,19,11,20 X6,15,7,16 X16,5,17,6 X4,17,5,18 X8,22,9,21 X2,11,3,12 X10,13,1,14
Gauss code	{1, -10, 3, -8, 7, -6, 2, -9, 4, -11}, {10, -1, 11, -3, 6, -7, 8, -2, 5, -4, 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+v-1) (u v-u-v) \left(u^2 v+u v^2-u v+u+v\right)}{u^2 v^2} }[/math] (db)
Jones polynomial	[math]\displaystyle{ \frac{15}{q^{9/2}}-\frac{13}{q^{7/2}}+\frac{10}{q^{5/2}}-\frac{7}{q^{3/2}}+\frac{1}{q^{21/2}}-\frac{2}{q^{19/2}}+\frac{4}{q^{17/2}}-\frac{9}{q^{15/2}}+\frac{11}{q^{13/2}}-\frac{14}{q^{11/2}}-\sqrt{q}+\frac{3}{\sqrt{q}} }[/math] (db)
Signature	-3 (db)
HOMFLY-PT polynomial	[math]\displaystyle{ a^9 \left(-z^3\right)-2 a^9 z+a^7 z^5+2 a^7 z^3+2 a^7 z+a^7 z^{-1} +2 a^5 z^5+4 a^5 z^3+a^5 z-a^5 z^{-1} +a^3 z^5-3 a^3 z-a z^3-a z }[/math] (db)
Kauffman polynomial	[math]\displaystyle{ a^{12} z^6-4 a^{12} z^4+4 a^{12} z^2+2 a^{11} z^7-7 a^{11} z^5+8 a^{11} z^3-4 a^{11} z+2 a^{10} z^8-3 a^{10} z^6-3 a^{10} z^4+4 a^{10} z^2+2 a^9 z^9-3 a^9 z^7+a^9 z^5+2 a^9 z^3-6 a^9 z+a^8 z^{10}+a^8 z^8-a^8 z^6-4 a^8 z^4+4 a^8 z^2+5 a^7 z^9-11 a^7 z^7+15 a^7 z^5-11 a^7 z^3+4 a^7 z-a^7 z^{-1} +a^6 z^{10}+4 a^6 z^8-8 a^6 z^6+5 a^6 z^4+a^6 z^2+a^6+3 a^5 z^9-a^5 z^7-3 a^5 z^5+2 a^5 z^3+2 a^5 z-a^5 z^{-1} +5 a^4 z^8-8 a^4 z^6+5 a^4 z^4-2 a^4 z^2+5 a^3 z^7-9 a^3 z^5+5 a^3 z^3-3 a^3 z+3 a^2 z^6-5 a^2 z^4+a^2 z^2+a z^5-2 a z^3+a z }[/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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 2                       1 1 0                     2   -2 -2                   5 1   4 -4                 6 3     -3 -6               7 4       3 -8             8 6         -2 -10           6 7           -1 -12         5 8             3 -14       4 6               -2 -16     1 6                 5 -18   1 3                   -2 -20   1                     1 -22 1                       -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-4 }[/math] [math]\displaystyle{ i=-2 }[/math] [math]\displaystyle{ r=-9 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-8 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-7 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-6 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{4} }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} }[/math] [math]\displaystyle{ {\mathbb Z}^{7} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=2 }[/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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