L11a412

L11a411

L11a413

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a412 at Knotilus!

Link Presentations

[edit Notes on L11a412's Link Presentations]

Planar diagram presentation	X6172 X12,3,13,4 X18,13,19,14 X22,17,11,18 X16,7,17,8 X8,22,9,21 X14,10,15,9 X20,16,21,15 X10,19,5,20 X2536 X4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, 5, -6, 7, -9}, {11, -2, 3, -7, 8, -5, 4, -3, 9, -8, 6, -4}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {uv^{2}w^{3}-3uv^{2}w^{2}+3uv^{2}w-uv^{2}+uvw^{4}-5uvw^{3}+9uvw^{2}-6uvw+2uv-uw^{4}+4uw^{3}-5uw^{2}+3uw-u+v^{2}w^{4}-3v^{2}w^{3}+5v^{2}w^{2}-4v^{2}w+v^{2}-2vw^{4}+6vw^{3}-9vw^{2}+5vw-v+w^{4}-3w^{3}+3w^{2}-w}{{\sqrt {u}}vw^{2}}}$ (db)
Jones polynomial	$-q^{2}+5q-12+20q^{-1}-25q^{-2}+31q^{-3}-28q^{-4}+25q^{-5}-18q^{-6}+10q^{-7}-4q^{-8}+q^{-9}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^{8}z^{2}+a^{8}-3a^{6}z^{4}-5a^{6}z^{2}+a^{6}z^{-2}-3a^{6}+2a^{4}z^{6}+5a^{4}z^{4}+6a^{4}z^{2}-2a^{4}z^{-2}+a^{4}+a^{2}z^{6}-a^{2}z^{2}+a^{2}z^{-2}+a^{2}-z^{4}$ (db)
Kauffman polynomial	$a^{10}z^{6}-2a^{10}z^{4}+a^{10}z^{2}+4a^{9}z^{7}-8a^{9}z^{5}+6a^{9}z^{3}-2a^{9}z+8a^{8}z^{8}-16a^{8}z^{6}+13a^{8}z^{4}-7a^{8}z^{2}+2a^{8}+8a^{7}z^{9}-6a^{7}z^{7}-14a^{7}z^{5}+18a^{7}z^{3}-7a^{7}z+3a^{6}z^{10}+21a^{6}z^{8}-64a^{6}z^{6}+59a^{6}z^{4}-24a^{6}z^{2}+a^{6}z^{-2}+4a^{6}+20a^{5}z^{9}-26a^{5}z^{7}-13a^{5}z^{5}+25a^{5}z^{3}-7a^{5}z-2a^{5}z^{-1}+3a^{4}z^{10}+30a^{4}z^{8}-79a^{4}z^{6}+63a^{4}z^{4}-21a^{4}z^{2}+2a^{4}z^{-2}+3a^{4}+12a^{3}z^{9}-4a^{3}z^{7}-22a^{3}z^{5}+17a^{3}z^{3}-3a^{3}z-2a^{3}z^{-1}+17a^{2}z^{8}-27a^{2}z^{6}+16a^{2}z^{4}-5a^{2}z^{2}+a^{2}z^{-2}+12az^{7}-14az^{5}+z^{5}a^{-1}+4az^{3}-az+5z^{6}-3z^{4}$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 5                       1 -1 3                     4   4 1                   8 1   -7 -1                 12 4     8 -3               14 9       -5 -5             17 11         6 -7           13 16           3 -9         12 15             -3 -11       7 14               7 -13     3 11                 -8 -15   1 7                   6 -17   3                     -3 -19 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-3$ $i=-1$ $r=-8$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-5$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-4$ ${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{12}$ $r=-3$ ${\mathbb {Z} }^{15}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{13}$ $r=-2$ ${\mathbb {Z} }^{16}\oplus {\mathbb {Z} }_{2}^{15}$ ${\mathbb {Z} }^{17}$ $r=-1$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{14}$ ${\mathbb {Z} }^{14}$ $r=0$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{12}$ $r=1$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=2$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=3$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

L11a411 L11a413