L11n233

L11n232

L11n234

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n233 at Knotilus!

Link Presentations

[edit Notes on L11n233's Link Presentations]

Planar diagram presentation	X10,1,11,2 X2,11,3,12 X12,3,13,4 X9,18,10,19 X6,13,7,14 X14,7,15,8 X8,15,1,16 X17,22,18,9 X21,16,22,17 X19,4,20,5 X5,20,6,21
Gauss code	{1, -2, 3, 10, -11, -5, 6, -7}, {-4, -1, 2, -3, 5, -6, 7, 9, -8, 4, -10, 11, -9, 8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$\frac{-u^3 v^5+u^2 v^4-3 u^2 v^3+2 u^2 v^2-u^2 v-u v^4+2 u v^3-3 u v^2+u v-1}{u^{3/2} v^{5/2}}$ (db)
Jones polynomial	$-\frac{1}{q^{9/2}}-\frac{1}{q^{13/2}}-\frac{1}{q^{15/2}}+\frac{3}{q^{17/2}}-\frac{4}{q^{19/2}}+\frac{5}{q^{21/2}}-\frac{5}{q^{23/2}}+\frac{4}{q^{25/2}}-\frac{3}{q^{27/2}}+\frac{1}{q^{29/2}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$-2 a^{13} z^3-4 a^{13} z-a^{13} z^{-1} +a^{11} z^7+9 a^{11} z^5+23 a^{11} z^3+19 a^{11} z+3 a^{11} z^{-1} -a^9 z^9-9 a^9 z^7-28 a^9 z^5-37 a^9 z^3-19 a^9 z-2 a^9 z^{-1}$ (db)
Kauffman polynomial	$a^{18} z^4-a^{18} z^2+3 a^{17} z^5-5 a^{17} z^3+a^{17} z+3 a^{16} z^6-4 a^{16} z^4+a^{15} z^7+3 a^{15} z^5-7 a^{15} z^3+3 a^{15} z+3 a^{14} z^6-2 a^{14} z^4-a^{14} z^2+a^{14}+4 a^{13} z^5-6 a^{13} z^3+5 a^{13} z-a^{13} z^{-1} +a^{12} z^8-9 a^{12} z^6+26 a^{12} z^4-21 a^{12} z^2+3 a^{12}+a^{11} z^9-10 a^{11} z^7+32 a^{11} z^5-41 a^{11} z^3+22 a^{11} z-3 a^{11} z^{-1} +a^{10} z^8-9 a^{10} z^6+23 a^{10} z^4-19 a^{10} z^2+3 a^{10}+a^9 z^9-9 a^9 z^7+28 a^9 z^5-37 a^9 z^3+19 a^9 z-2 a^9 z^{-1}$ (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   \ -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -8                       1 1 -10                       1 1 -12                   1     1 -14               2         2 -16             2 1 1       -2 -18           3 2           1 -20         2 2 1           -1 -22       3 3               0 -24     2 3                 1 -26   1 2                   -1 -28   2                     2 -30 1                       -1

Integral Khovanov Homology (db, data source)

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-10$ $i=-8$ $i=-6$ $r=-11$ ${\mathbb Z}$ $r=-10$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-9$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-8$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=-7$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-6$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-5$ ${\mathbb Z}$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-4$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=-3$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

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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