L11n345

L11n344

L11n346

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n345 at Knotilus!

Link Presentations

[edit Notes on L11n345's Link Presentations]

Planar diagram presentation	X6172 X14,4,15,3 X11,19,12,18 X16,8,17,7 X17,21,18,20 X19,5,20,12 X8,22,9,21 X22,10,13,9 X10,14,11,13 X2536 X4,16,1,15
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -7, 8, -9, -3, 6}, {9, -2, 11, -4, -5, 3, -6, 5, 7, -8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(1)t(3)^{3}t(2)^{3}-t(3)^{3}t(2)^{3}+t(3)^{2}t(2)^{3}-t(1)t(3)^{3}t(2)^{2}+t(1)t(3)t(2)^{2}-t(3)^{2}t(2)+t(2)+t(1)-t(1)t(3)-1}{{\sqrt {t(1)}}t(2)^{3/2}t(3)^{3/2}}}$ (db)
Jones polynomial	$q^{10}-q^{8}+q^{7}-q^{6}+3q^{5}-2q^{4}+3q^{3}-q^{2}+q$ (db)
Signature	6 (db)
HOMFLY-PT polynomial	$-z^{8}a^{-6}+z^{6}a^{-4}-7z^{6}a^{-6}+z^{6}a^{-8}+6z^{4}a^{-4}-17z^{4}a^{-6}+6z^{4}a^{-8}+11z^{2}a^{-4}-20z^{2}a^{-6}+9z^{2}a^{-8}-z^{2}a^{-10}+7a^{-4}-11a^{-6}+4a^{-8}+a^{-4}z^{-2}-2a^{-6}z^{-2}+a^{-8}z^{-2}$ (db)
Kauffman polynomial	$z^{4}a^{-12}-4z^{2}a^{-12}+a^{-12}-z^{3}a^{-11}-z^{4}a^{-10}-z^{2}a^{-10}+2z^{7}a^{-9}-11z^{5}a^{-9}+11z^{3}a^{-9}+3z^{8}a^{-8}-18z^{6}a^{-8}+29z^{4}a^{-8}-16z^{2}a^{-8}-a^{-8}z^{-2}+5a^{-8}+z^{9}a^{-7}-3z^{7}a^{-7}-7z^{5}a^{-7}+20z^{3}a^{-7}-11za^{-7}+2a^{-7}z^{-1}+4z^{8}a^{-6}-25z^{6}a^{-6}+48z^{4}a^{-6}-37z^{2}a^{-6}-2a^{-6}z^{-2}+13a^{-6}+z^{9}a^{-5}-5z^{7}a^{-5}+4z^{5}a^{-5}+8z^{3}a^{-5}-11za^{-5}+2a^{-5}z^{-1}+z^{8}a^{-4}-7z^{6}a^{-4}+17z^{4}a^{-4}-18z^{2}a^{-4}-a^{-4}z^{-2}+8a^{-4}$ (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 8 χ 21                     1 1 19                     1 1 17               2 1     -1 15             1 1       0 13           2 3 1       0 11         1 1 2         2 9       1 2             1 7     2 1               1 5   1 3                 2 3                       0 1 1                     1

Integral Khovanov Homology (db, data source)

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

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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