L11n370

L11n369

L11n371

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n370 at Knotilus!

Link Presentations

[edit Notes on L11n370's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X14,8,15,7 X15,20,16,21 X11,19,12,18 X17,13,18,12 X19,22,20,17 X21,16,22,5 X8,14,9,13 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {-6, 5, -7, 4, -8, 7}, {10, -1, 3, -9, 11, -2, -5, 6, 9, -3, -4, 8}

A Braid Representative
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	$-q^{-6}+5q^{-5}-7q^{-4}+q^{3}+11q^{-3}-3q^{2}-10q^{-2}+6q+11q^{-1}-9$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^{6}z^{-2}-a^{4}z^{4}-a^{4}z^{2}-2a^{4}z^{-2}-2a^{4}+a^{2}z^{6}+3a^{2}z^{4}+4a^{2}z^{2}+a^{2}z^{-2}+z^{2}a^{-2}+3a^{2}+a^{-2}-2z^{4}-4z^{2}-2$ (db)
Kauffman polynomial	$a^{7}z^{3}+5a^{6}z^{4}-a^{6}z^{2}+a^{6}z^{-2}-2a^{6}+2a^{5}z^{7}+2a^{5}z^{3}+2a^{5}z-2a^{5}z^{-1}+3a^{4}z^{8}-4a^{4}z^{6}+6a^{4}z^{4}-3a^{4}z^{2}+2a^{4}z^{-2}-2a^{4}+a^{3}z^{9}+6a^{3}z^{7}-16a^{3}z^{5}+12a^{3}z^{3}-2a^{3}z-2a^{3}z^{-1}+6a^{2}z^{8}-9a^{2}z^{6}+z^{6}a^{-2}-3a^{2}z^{4}-3z^{4}a^{-2}+3a^{2}z^{2}+3z^{2}a^{-2}+a^{2}z^{-2}-a^{2}-a^{-2}+az^{9}+7az^{7}+3z^{7}a^{-1}-25az^{5}-9z^{5}a^{-1}+19az^{3}+8z^{3}a^{-1}-6az-2za^{-1}+3z^{8}-4z^{6}-7z^{4}+8z^{2}-1$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 χ 7                   1 1 5                 2   -2 3               4 1   3 1             5 2     -3 -1           6 4       2 -3         6 7         1 -5       5 4           1 -7     2 6             4 -9   3 5               -2 -11   4                 4 -13 1                   -1

Integral Khovanov Homology (db, data source)

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

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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