L11n264

L11n263

L11n265

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n264 at Knotilus!

Link Presentations

[edit Notes on L11n264's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X14,7,15,8 X8,13,5,14 X18,12,19,11 X19,22,20,9 X15,20,16,21 X21,16,22,17 X12,18,13,17 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -4}, {11, -2, 5, -9, 4, -3, -7, 8, 9, -5, -6, 7, -8, 6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {-t(1)t(3)^{3}+t(1)t(2)t(3)^{3}-t(2)t(3)^{3}+t(3)^{3}+t(1)t(3)^{2}-2t(1)t(2)t(3)^{2}+t(2)t(3)^{2}-t(3)^{2}-t(1)t(3)+t(1)t(2)t(3)-t(2)t(3)+2t(3)+t(1)-t(1)t(2)+t(2)-1}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{3/2}}}$ (db)
Jones polynomial	$1-3q^{-1}+5q^{-2}-5q^{-3}+7q^{-4}-5q^{-5}+6q^{-6}-3q^{-7}+q^{-8}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$a^{8}z^{-2}+z^{4}a^{6}+z^{2}a^{6}-2a^{6}z^{-2}-3a^{6}-z^{6}a^{4}-3z^{4}a^{4}+a^{4}z^{-2}+3a^{4}+z^{4}a^{2}+2z^{2}a^{2}$ (db)
Kauffman polynomial	$2a^{9}z^{3}+a^{8}z^{8}-5a^{8}z^{6}+11a^{8}z^{4}-6a^{8}z^{2}-a^{8}z^{-2}+3a^{8}+a^{7}z^{9}-3a^{7}z^{7}+2a^{7}z^{5}+3a^{7}z^{3}-3a^{7}z+2a^{7}z^{-1}+4a^{6}z^{8}-16a^{6}z^{6}+22a^{6}z^{4}-14a^{6}z^{2}-2a^{6}z^{-2}+5a^{6}+a^{5}z^{9}-8a^{5}z^{5}+7a^{5}z^{3}-3a^{5}z+2a^{5}z^{-1}+3a^{4}z^{8}-10a^{4}z^{6}+8a^{4}z^{4}-6a^{4}z^{2}-a^{4}z^{-2}+3a^{4}+3a^{3}z^{7}-10a^{3}z^{5}+6a^{3}z^{3}+a^{2}z^{6}-3a^{2}z^{4}+2a^{2}z^{2}$ (db)

Khovanov Homology

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

\   r    \    j   \ -6 -5 -4 -3 -2 -1 0 1 2 χ 1                 1 1 -1               2   -2 -3             3 1   2 -5           3 3     0 -7         4 2       2 -9     1 2 3         2 -11     5 4           1 -13   1 4             3 -15   2               -2 -17 1                 1

Integral Khovanov Homology (db, data source)

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

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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