L11n412

L11n411

L11n413

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n412 at Knotilus!

Link Presentations

[edit Notes on L11n412's Link Presentations]

Planar diagram presentation	X8192 X5,15,6,14 X10,3,11,4 X13,5,14,4 X2738 X6,9,1,10 X11,18,12,19 X17,12,18,7 X15,20,16,21 X19,22,20,13 X21,16,22,17
Gauss code	{1, -5, 3, 4, -2, -6}, {5, -1, 6, -3, -7, 8}, {-4, 2, -9, 11, -8, 7, -10, 9, -11, 10}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(1)^{2}t(3)^{3}-2t(1)^{2}t(3)^{2}+t(1)t(2)^{2}t(3)^{2}-2t(2)^{2}t(3)^{2}+t(1)t(3)^{2}+t(1)^{2}t(2)t(3)^{2}-2t(1)t(2)t(3)^{2}+t(2)t(3)^{2}+2t(1)^{2}t(3)-t(1)t(2)^{2}t(3)+2t(2)^{2}t(3)-t(1)t(3)-t(1)^{2}t(2)t(3)+2t(1)t(2)t(3)-t(2)t(3)-t(2)^{2}}{t(1)t(2)t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{-10}+2q^{-9}-4q^{-8}+6q^{-7}-6q^{-6}+8q^{-5}-6q^{-4}+6q^{-3}-3q^{-2}+2q^{-1}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-a^{10}z^{-2}-a^{10}+3z^{2}a^{8}+4a^{8}z^{-2}+6a^{8}-2z^{4}a^{6}-6z^{2}a^{6}-5a^{6}z^{-2}-9a^{6}-z^{4}a^{4}+2a^{4}z^{-2}+2a^{4}+2z^{2}a^{2}+2a^{2}$ (db)
Kauffman polynomial	$a^{11}z^{7}-5a^{11}z^{5}+8a^{11}z^{3}-5a^{11}z+a^{11}z^{-1}+2a^{10}z^{8}-9a^{10}z^{6}+12a^{10}z^{4}-7a^{10}z^{2}-a^{10}z^{-2}+4a^{10}+a^{9}z^{9}+a^{9}z^{7}-20a^{9}z^{5}+35a^{9}z^{3}-21a^{9}z+5a^{9}z^{-1}+6a^{8}z^{8}-26a^{8}z^{6}+38a^{8}z^{4}-32a^{8}z^{2}-4a^{8}z^{-2}+17a^{8}+a^{7}z^{9}+4a^{7}z^{7}-28a^{7}z^{5}+44a^{7}z^{3}-33a^{7}z+9a^{7}z^{-1}+4a^{6}z^{8}-15a^{6}z^{6}+26a^{6}z^{4}-32a^{6}z^{2}-5a^{6}z^{-2}+20a^{6}+4a^{5}z^{7}-12a^{5}z^{5}+18a^{5}z^{3}-16a^{5}z+5a^{5}z^{-1}+2a^{4}z^{6}-4a^{4}z^{2}-2a^{4}z^{-2}+6a^{4}+a^{3}z^{5}+a^{3}z^{3}+a^{3}z+3a^{2}z^{2}-2a^{2}$ (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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -1                   2 2 -3                 3 2 -1 -5               3     3 -7             4 4     0 -9           4 2       2 -11         2 4         2 -13       4 4           0 -15     1 3             2 -17   1 3               -2 -19   1                 1 -21 1                   -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-3$ $i=-1$ $r=-9$ ${\mathbb {Z} }$ $r=-8$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{4}$ $r=-5$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-4$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-3$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-2$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{3}$ $r=-1$ ${\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=0$ ${\mathbb {Z} }^{2}$ ${\mathbb {Z} }^{2}$

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.

L11n411 L11n413