L11n56

L11n55

L11n57

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n56 at Knotilus!

Link Presentations

[edit Notes on L11n56's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X11,17,12,16 X14,7,15,8 X8,15,9,16 X13,21,14,20 X17,5,18,22 X21,19,22,18 X19,13,20,12 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -5, 11, -2, -3, 9, -6, -4, 5, 3, -7, 8, -9, 6, -8, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{2 t(1) t(2)^3-t(2)^3-4 t(1) t(2)^2+4 t(2)^2+4 t(1) t(2)-4 t(2)-t(1)+2}{\sqrt{t(1)} t(2)^{3/2}}$ (db)
Jones polynomial	$q^{9/2}-\frac{1}{q^{9/2}}-3 q^{7/2}+\frac{2}{q^{7/2}}+5 q^{5/2}-\frac{5}{q^{5/2}}-6 q^{3/2}+\frac{6}{q^{3/2}}+7 \sqrt{q}-\frac{8}{\sqrt{q}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^3 z^3+z^3 a^{-3} +2 a^3 z+z a^{-3} +2 a^3 z^{-1} -a z^5-z^5 a^{-1} -3 a z^3-2 z^3 a^{-1} -5 a z-3 a z^{-1} + a^{-1} z^{-1}$ (db)
Kauffman polynomial	$-a z^9-z^9 a^{-1} -a^2 z^8-3 z^8 a^{-2} -4 z^8+2 a z^7-z^7 a^{-1} -3 z^7 a^{-3} +2 a^2 z^6+9 z^6 a^{-2} -z^6 a^{-4} +12 z^6-4 a^3 z^5-5 a z^5+9 z^5 a^{-1} +10 z^5 a^{-3} -2 a^4 z^4-7 a^2 z^4-5 z^4 a^{-2} +3 z^4 a^{-4} -13 z^4-a^5 z^3+6 a^3 z^3+9 a z^3-5 z^3 a^{-1} -7 z^3 a^{-3} +a^4 z^2+7 a^2 z^2+z^2 a^{-2} -2 z^2 a^{-4} +9 z^2+a^5 z-6 a^3 z-9 a z-z a^{-1} +z a^{-3} -3 a^2- a^{-2} -3+2 a^3 z^{-1} +3 a z^{-1} + a^{-1} z^{-1}$ (db)

Khovanov Homology

The coefficients of the monomials $t^rq^j$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$).

\   r    \    j   \ -4 -3 -2 -1 0 1 2 3 4 5 χ 10                   1 -1 8                 2   2 6               3 1   -2 4             3 2     1 2           4 3       -1 0         4 3         1 -2       3 5           2 -4     2 3             -1 -6     3               3 -8 1 2                 -1 -10 1                   1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-4$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ $r=1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

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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