L11n362

L11n361

L11n363

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n362 at Knotilus!

Link Presentations

[edit Notes on L11n362's Link Presentations]

Planar diagram presentation	X6172 X12,7,13,8 X4,13,1,14 X18,6,19,5 X8493 X9,21,10,20 X19,11,20,10 X14,18,15,17 X22,16,17,15 X16,22,5,21 X2,12,3,11
Gauss code	{1, -11, 5, -3}, {8, -4, -7, 6, 10, -9}, {4, -1, 2, -5, -6, 7, 11, -2, 3, -8, 9, -10}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(3)-1)^{2}\left(t(3)t(2)^{2}+t(3)^{2}t(2)-t(3)t(2)+t(2)+t(3)\right)}{{\sqrt {t(1)}}t(2)t(3)^{2}}}$ (db)
Jones polynomial	$3q^{7}-5q^{6}+10q^{5}-12q^{4}+14q^{3}-13q^{2}-q^{-2}+11q+4q^{-1}-7$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{6}a^{-2}+z^{6}a^{-4}+2z^{4}a^{-2}+3z^{4}a^{-4}-z^{4}a^{-6}-z^{4}+4z^{2}a^{-4}-3z^{2}a^{-6}-z^{2}-a^{-2}+3a^{-4}-4a^{-6}+a^{-8}+1+a^{-4}z^{-2}-2a^{-6}z^{-2}+a^{-8}z^{-2}$ (db)
Kauffman polynomial	$6z^{4}a^{-8}-14z^{2}a^{-8}-a^{-8}z^{-2}+8a^{-8}+3z^{7}a^{-7}-6z^{5}a^{-7}+10z^{3}a^{-7}-10za^{-7}+2a^{-7}z^{-1}+5z^{8}a^{-6}-16z^{6}a^{-6}+33z^{4}a^{-6}-33z^{2}a^{-6}-2a^{-6}z^{-2}+15a^{-6}+2z^{9}a^{-5}+3z^{7}a^{-5}-13z^{5}a^{-5}+20z^{3}a^{-5}-12za^{-5}+2a^{-5}z^{-1}+10z^{8}a^{-4}-25z^{6}a^{-4}+33z^{4}a^{-4}-24z^{2}a^{-4}-a^{-4}z^{-2}+9a^{-4}+2z^{9}a^{-3}+6z^{7}a^{-3}-18z^{5}a^{-3}+14z^{3}a^{-3}-3za^{-3}+5z^{8}a^{-2}-5z^{6}a^{-2}-z^{4}a^{-2}-3z^{2}a^{-2}+2a^{-2}+6z^{7}a^{-1}+az^{5}-10z^{5}a^{-1}-az^{3}+3z^{3}a^{-1}-za^{-1}+4z^{6}-7z^{4}+2z^{2}+1$ (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   \ -3 -2 -1 0 1 2 3 4 5 6 χ 15                   3 3 13                 4 2 -2 11               6 1   5 9             6 4     -2 7           8 6       2 5         5 6         1 3       6 8           -2 1     3 7             4 -1   1 4               -3 -3   3                 3 -5 1                   -1

Integral Khovanov Homology (db, data source)

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

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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