L11n364

L11n363

L11n365

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n364 at Knotilus!

Link Presentations

[edit Notes on L11n364's Link Presentations]

Planar diagram presentation	X6172 X10,4,11,3 X12,8,13,7 X15,17,16,22 X9,18,10,19 X17,8,18,9 X13,21,14,20 X21,15,22,14 X19,5,20,16 X2536 X4,12,1,11
Gauss code	{1, -10, 2, -11}, {-6, 5, -9, 7, -8, 4}, {10, -1, 3, 6, -5, -2, 11, -3, -7, 8, -4, 9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(u-1)(w-1)\left(v^{2}w-2vw+2v-1\right)}{{\sqrt {u}}vw}}$ (db)
Jones polynomial	$-q^{8}+3q^{7}-5q^{6}+7q^{5}-8q^{4}+9q^{3}-6q^{2}+6q+q^{-1}-2$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-z^{4}a^{-6}-2z^{2}a^{-6}-a^{-6}+z^{6}a^{-4}+4z^{4}a^{-4}+6z^{2}a^{-4}+a^{-4}z^{-2}+3a^{-4}-2z^{4}a^{-2}-5z^{2}a^{-2}-2a^{-2}z^{-2}-4a^{-2}+z^{2}+z^{-2}+2$ (db)
Kauffman polynomial	$z^{5}a^{-9}-2z^{3}a^{-9}+3z^{6}a^{-8}-7z^{4}a^{-8}+2z^{2}a^{-8}+4z^{7}a^{-7}-10z^{5}a^{-7}+6z^{3}a^{-7}-za^{-7}+3z^{8}a^{-6}-7z^{6}a^{-6}+7z^{4}a^{-6}-5z^{2}a^{-6}+2a^{-6}+z^{9}a^{-5}+z^{7}a^{-5}-6z^{5}a^{-5}+9z^{3}a^{-5}-3za^{-5}+4z^{8}a^{-4}-14z^{6}a^{-4}+27z^{4}a^{-4}-21z^{2}a^{-4}-a^{-4}z^{-2}+7a^{-4}+z^{9}a^{-3}-3z^{7}a^{-3}+7z^{5}a^{-3}-z^{3}a^{-3}-4za^{-3}+2a^{-3}z^{-1}+z^{8}a^{-2}-4z^{6}a^{-2}+14z^{4}a^{-2}-17z^{2}a^{-2}-2a^{-2}z^{-2}+7a^{-2}+2z^{5}a^{-1}-2z^{3}a^{-1}-2za^{-1}+2a^{-1}z^{-1}+z^{4}-3z^{2}-z^{-2}+3$ (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   \ -2 -1 0 1 2 3 4 5 6 7 χ 17                   1 -1 15                 2   2 13               3 1   -2 11             4 2     2 9           4 3       -1 7         5 4         1 5       3 6           3 3     3 3             0 1   1 5               4 -1   1                 -1 -3 1                   1

Integral Khovanov Homology (db, data source)

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

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.

L11n363 L11n365