L11n288

L11n287

L11n289

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n288 at Knotilus!

Link Presentations

[edit Notes on L11n288's Link Presentations]

Planar diagram presentation	X6172 X5,12,6,13 X3849 X13,2,14,3 X14,7,15,8 X11,18,12,19 X9,21,10,20 X19,5,20,10 X4,15,1,16 X17,22,18,11 X21,16,22,17
Gauss code	{1, 4, -3, -9}, {-2, -1, 5, 3, -7, 8}, {-6, 2, -4, -5, 9, 11, -10, 6, -8, 7, -11, 10}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-t(1)t(2)^{2}t(3)^{4}+t(1)t(2)t(3)^{4}+t(1)t(2)^{2}t(3)^{3}-t(3)^{3}-t(1)t(2)^{2}t(3)^{2}+t(3)^{2}+t(1)t(2)^{2}t(3)-t(3)-t(2)+1}{{\sqrt {t(1)}}t(2)t(3)^{2}}}$ (db)
Jones polynomial	$-q^{-10}+q^{-9}-2q^{-8}+3q^{-7}-2q^{-6}+4q^{-5}-2q^{-4}+3q^{-3}-q^{-2}+q^{-1}$ (db)
Signature	-6 (db)
HOMFLY-PT polynomial	$-z^{2}a^{10}-a^{10}z^{-2}-2a^{10}+z^{6}a^{8}+6z^{4}a^{8}+11z^{2}a^{8}+4a^{8}z^{-2}+9a^{8}-z^{8}a^{6}-7z^{6}a^{6}-17z^{4}a^{6}-20z^{2}a^{6}-5a^{6}z^{-2}-14a^{6}+z^{6}a^{4}+6z^{4}a^{4}+11z^{2}a^{4}+2a^{4}z^{-2}+7a^{4}$ (db)
Kauffman polynomial	$a^{13}z+a^{12}z^{2}+2a^{11}z^{3}-4a^{11}z+a^{11}z^{-1}+a^{10}z^{6}-2a^{10}z^{4}-4a^{10}z^{2}-a^{10}z^{-2}+5a^{10}+3a^{9}z^{7}-16a^{9}z^{5}+27a^{9}z^{3}-21a^{9}z+5a^{9}z^{-1}+3a^{8}z^{8}-17a^{8}z^{6}+32a^{8}z^{4}-32a^{8}z^{2}-4a^{8}z^{-2}+18a^{8}+a^{7}z^{9}-2a^{7}z^{7}-12a^{7}z^{5}+34a^{7}z^{3}-29a^{7}z+9a^{7}z^{-1}+4a^{6}z^{8}-25a^{6}z^{6}+51a^{6}z^{4}-45a^{6}z^{2}-5a^{6}z^{-2}+21a^{6}+a^{5}z^{9}-5a^{5}z^{7}+4a^{5}z^{5}+9a^{5}z^{3}-13a^{5}z+5a^{5}z^{-1}+a^{4}z^{8}-7a^{4}z^{6}+17a^{4}z^{4}-18a^{4}z^{2}-2a^{4}z^{-2}+9a^{4}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 χ -1                   1 1 -3                     0 -5               3 1   2 -7           1 1 1     1 -9           4 2       2 -11       2 2 2         2 -13       3 2           1 -15   1 2 2             1 -17   1 2               -1 -19 1 1                 0 -21 1                   -1

Integral Khovanov Homology (db, data source)

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

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.

L11n287 L11n289