L11n207

L11n206

L11n208

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n207 at Knotilus!

Link Presentations

[edit Notes on L11n207's Link Presentations]

Planar diagram presentation	X10,1,11,2 X2,11,3,12 X12,3,13,4 X5,14,6,15 X17,22,18,9 X19,5,20,4 X21,6,22,7 X7,17,8,16 X8,9,1,10 X13,18,14,19 X15,21,16,20
Gauss code	{1, -2, 3, 6, -4, 7, -8, -9}, {9, -1, 2, -3, -10, 4, -11, 8, -5, 10, -6, 11, -7, 5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-\frac{u^3 v^3-u^3 v^2-2 u^2 v^3+6 u^2 v^2-4 u^2 v+u^2+u v^3-4 u v^2+6 u v-2 u-v+1}{u^{3/2} v^{3/2}}$ (db)
Jones polynomial	$\frac{10}{q^{9/2}}-\frac{10}{q^{7/2}}+\frac{6}{q^{5/2}}-\frac{4}{q^{3/2}}-\frac{1}{q^{19/2}}+\frac{3}{q^{17/2}}-\frac{6}{q^{15/2}}+\frac{9}{q^{13/2}}-\frac{10}{q^{11/2}}+\frac{1}{\sqrt{q}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$z^5 a^7+3 z^3 a^7+3 z a^7-z^7 a^5-5 z^5 a^5-9 z^3 a^5-5 z a^5+a^5 z^{-1} +z^5 a^3+2 z^3 a^3-a^3 z^{-1}$ (db)
Kauffman polynomial	$-z^5 a^{11}+2 z^3 a^{11}-z a^{11}-3 z^6 a^{10}+6 z^4 a^{10}-3 z^2 a^{10}-4 z^7 a^9+6 z^5 a^9-z a^9-3 z^8 a^8+2 z^6 a^8+2 z^4 a^8+z^2 a^8-z^9 a^7-3 z^7 a^7+2 z^5 a^7+6 z^3 a^7-4 z a^7-4 z^8 a^6+5 z^6 a^6-6 z^4 a^6+5 z^2 a^6-z^9 a^5+z^7 a^5-9 z^5 a^5+13 z^3 a^5-5 z a^5-a^5 z^{-1} -z^8 a^4-3 z^4 a^4+2 z^2 a^4+a^4-4 z^5 a^3+5 z^3 a^3-z a^3-a^3 z^{-1} -z^4 a^2+z^2 a^2$ (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 0                   1 -1 -2                 3   3 -4               4 2   -2 -6             6 2     4 -8           4 4       0 -10         6 6         0 -12       4 5           1 -14     2 5             -3 -16   1 4               3 -18   2                 -2 -20 1                   1

Integral Khovanov Homology (db, data source)

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

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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