L11n353

L11n352

L11n354

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n353 at Knotilus!

Link Presentations

[edit Notes on L11n353's Link Presentations]

Planar diagram presentation	X6172 X12,7,13,8 X4,13,1,14 X5,16,6,17 X8493 X9,21,10,20 X19,11,20,10 X17,22,18,15 X21,18,22,19 X15,14,16,5 X2,12,3,11
Gauss code	{1, -11, 5, -3}, {-10, 4, -8, 9, -7, 6, -9, 8}, {-4, -1, 2, -5, -6, 7, 11, -2, 3, 10}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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