L11n274

L11n273

L11n275

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n274 at Knotilus!

Link Presentations

[edit Notes on L11n274's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {uv^{2}w^{4}-2uv^{2}w^{3}+2uv^{2}w^{2}+uvw^{3}-uvw^{2}+vw^{2}-vw-2w^{2}+2w-1}{{\sqrt {u}}vw^{2}}}$ (db)
Jones polynomial	$-q+2-3q^{-1}+4q^{-2}-4q^{-3}+5q^{-4}-3q^{-5}+4q^{-6}-q^{-7}+q^{-8}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$a^{8}z^{2}+2a^{8}z^{-2}+2a^{8}-a^{6}z^{6}-6a^{6}z^{4}-12a^{6}z^{2}-5a^{6}z^{-2}-12a^{6}+a^{4}z^{8}+7a^{4}z^{6}+18a^{4}z^{4}+23a^{4}z^{2}+4a^{4}z^{-2}+14a^{4}-a^{2}z^{6}-5a^{2}z^{4}-7a^{2}z^{2}-a^{2}z^{-2}-4a^{2}$ (db)
Kauffman polynomial	$a^{10}z^{2}-a^{10}+a^{9}z^{3}+3a^{8}z^{4}-8a^{8}z^{2}-2a^{8}z^{-2}+8a^{8}+2a^{7}z^{7}-10a^{7}z^{5}+20a^{7}z^{3}-18a^{7}z+5a^{7}z^{-1}+3a^{6}z^{8}-16a^{6}z^{6}+31a^{6}z^{4}-33a^{6}z^{2}-5a^{6}z^{-2}+20a^{6}+a^{5}z^{9}-19a^{5}z^{5}+41a^{5}z^{3}-33a^{5}z+9a^{5}z^{-1}+5a^{4}z^{8}-26a^{4}z^{6}+42a^{4}z^{4}-32a^{4}z^{2}-4a^{4}z^{-2}+15a^{4}+a^{3}z^{9}-a^{3}z^{7}-14a^{3}z^{5}+29a^{3}z^{3}-19a^{3}z+5a^{3}z^{-1}+2a^{2}z^{8}-10a^{2}z^{6}+14a^{2}z^{4}-8a^{2}z^{2}-a^{2}z^{-2}+3a^{2}+az^{7}-5az^{5}+7az^{3}-4az+az^{-1}$ (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 3 χ 3                   1 -1 1                 1   1 -1               2 1   -1 -3           1 3 1     1 -5           3 3       0 -7       1 4 2         1 -9     1 2 3           2 -11     4 3             1 -13 1   2               3 -15 2 2                 0 -17 1                   1

Integral Khovanov Homology (db, data source)

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

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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