L11n393

L11n392

L11n394

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n393 at Knotilus!

Link Presentations

[edit Notes on L11n393's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X4,17,1,18 X12,6,13,5 X8493 X13,22,14,19 X9,20,10,21 X19,10,20,11 X21,14,22,15 X18,12,5,11 X2,16,3,15
Gauss code	{1, -11, 5, -3}, {-8, 7, -9, 6}, {4, -1, 2, -5, -7, 8, 10, -4, -6, 9, 11, -2, 3, -10}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{(t(1)-1) (t(3)-1)^2 \left(t(3)^3+t(2)\right)}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{5/2}}$ (db)
Jones polynomial	$- q^{-7} +3 q^{-6} -4 q^{-5} +7 q^{-4} +q^3-5 q^{-3} -q^2+6 q^{-2} -q-4 q^{-1} +3$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^6 \left(-z^2\right)+a^6 z^{-2} -a^6+2 a^4 z^4+5 a^4 z^2-2 a^4 z^{-2} +a^4-a^2 z^6-4 a^2 z^4-4 a^2 z^2+a^2 z^{-2} +z^2 a^{-2} + a^{-2} -z^2-1$ (db)
Kauffman polynomial	$a^7 z^7-4 a^7 z^5+4 a^7 z^3-a^7 z+3 a^6 z^8-14 a^6 z^6+19 a^6 z^4-10 a^6 z^2+a^6 z^{-2} +2 a^5 z^9-5 a^5 z^7-7 a^5 z^5+14 a^5 z^3-3 a^5 z-2 a^5 z^{-1} +8 a^4 z^8-39 a^4 z^6+55 a^4 z^4-31 a^4 z^2+2 a^4 z^{-2} +4 a^4+2 a^3 z^9-4 a^3 z^7-15 a^3 z^5+30 a^3 z^3-11 a^3 z-2 a^3 z^{-1} +5 a^2 z^8-27 a^2 z^6+z^6 a^{-2} +41 a^2 z^4-5 z^4 a^{-2} -23 a^2 z^2+4 z^2 a^{-2} +a^2 z^{-2} +5 a^2- a^{-2} +3 a z^7+z^7 a^{-1} -19 a z^5-7 z^5 a^{-1} +30 a z^3+10 z^3 a^{-1} -13 a z-4 z a^{-1} -z^6+2 z^2+1$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 7                       1 1 5                         0 3                 2 1 1   -2 1               3 1       2 -1             4 4 1       -1 -3           3 3 2         2 -5         3 4             1 -7       4 3 1             2 -9     2 5                 3 -11   1 2                   -1 -13   2                     2 -15 1                       -1

Integral Khovanov Homology (db, data source)

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

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.

L11n392 L11n394