L11n117

L11n116

L11n118

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n117 at Knotilus!

Link Presentations

[edit Notes on L11n117's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X17,21,18,20 X13,19,14,18 X19,15,20,14 X4,21,1,22 X10,5,11,6 X12,3,13,4 X22,11,5,12 X2,9,3,10 X8,15,9,16
Gauss code	{1, -10, 8, -6}, {7, -1, 2, -11, 10, -7, 9, -8, -4, 5, 11, -2, -3, 4, -5, 3, 6, -9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{t(2)^5+2 t(1) t(2)^4-4 t(2)^4-3 t(1) t(2)^3+5 t(2)^3+5 t(1) t(2)^2-3 t(2)^2-4 t(1) t(2)+2 t(2)+t(1)}{\sqrt{t(1)} t(2)^{5/2}}$ (db)
Jones polynomial	$2 \sqrt{q}-\frac{5}{\sqrt{q}}+\frac{7}{q^{3/2}}-\frac{10}{q^{5/2}}+\frac{9}{q^{7/2}}-\frac{10}{q^{9/2}}+\frac{8}{q^{11/2}}-\frac{5}{q^{13/2}}+\frac{3}{q^{15/2}}-\frac{1}{q^{17/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$z^3 a^7+z a^7-a^7 z^{-1} -z^5 a^5-z^3 a^5+3 z a^5+4 a^5 z^{-1} -2 z^5 a^3-7 z^3 a^3-9 z a^3-4 a^3 z^{-1} +2 z^3 a+3 z a+a z^{-1}$ (db)
Kauffman polynomial	$-z^7 a^9+4 z^5 a^9-5 z^3 a^9+2 z a^9-3 z^8 a^8+13 z^6 a^8-17 z^4 a^8+6 z^2 a^8+a^8-2 z^9 a^7+3 z^7 a^7+10 z^5 a^7-16 z^3 a^7+5 z a^7-a^7 z^{-1} -9 z^8 a^6+32 z^6 a^6-28 z^4 a^6+3 z^2 a^6+4 a^6-2 z^9 a^5-4 z^7 a^5+30 z^5 a^5-31 z^3 a^5+13 z a^5-4 a^5 z^{-1} -6 z^8 a^4+14 z^6 a^4-2 z^4 a^4-10 z^2 a^4+7 a^4-8 z^7 a^3+23 z^5 a^3-25 z^3 a^3+14 z a^3-4 a^3 z^{-1} -5 z^6 a^2+9 z^4 a^2-10 z^2 a^2+4 a^2-z^5 a-5 z^3 a+4 z a-a z^{-1} -3 z^2+1$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 χ 2                   2 -2 0                 3   3 -2               5 3   -2 -4             5 2     3 -6           4 5       1 -8         6 5         1 -10       3 5           2 -12     2 5             -3 -14   1 3               2 -16   2                 -2 -18 1                   1

Integral Khovanov Homology (db, data source)

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

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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