L11a411

L11a410

L11a412

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a411 at Knotilus!

Link Presentations

[edit Notes on L11a411's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {2uv^{2}w^{2}-2uv^{2}w-4uvw^{2}+5uvw-2uv+2uw^{2}-4uw+2u-2v^{2}w^{2}+4v^{2}w-2v^{2}+2vw^{2}-5vw+4v+2w-2}{{\sqrt {u}}vw}}$ (db)
Jones polynomial	$-q^{8}+3q^{7}-6q^{6}+10q^{5}-13q^{4}+15q^{3}-14q^{2}+13q-8+6q^{-1}-2q^{-2}+q^{-3}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{6}a^{-2}+z^{6}a^{-4}+2z^{4}a^{-2}+3z^{4}a^{-4}-z^{4}a^{-6}-2z^{4}+a^{2}z^{2}+z^{2}a^{-2}+4z^{2}a^{-4}-2z^{2}a^{-6}-5z^{2}+2a^{2}+a^{-2}+2a^{-4}-a^{-6}-4+a^{2}z^{-2}+a^{-2}z^{-2}-2z^{-2}$ (db)
Kauffman polynomial	$z^{10}a^{-2}+z^{10}a^{-4}+2z^{9}a^{-1}+6z^{9}a^{-3}+4z^{9}a^{-5}+3z^{8}a^{-2}+7z^{8}a^{-4}+6z^{8}a^{-6}+2z^{8}+2az^{7}-z^{7}a^{-1}-15z^{7}a^{-3}-7z^{7}a^{-5}+5z^{7}a^{-7}+a^{2}z^{6}-9z^{6}a^{-2}-27z^{6}a^{-4}-16z^{6}a^{-6}+3z^{6}a^{-8}-5az^{5}-2z^{5}a^{-1}+18z^{5}a^{-3}+3z^{5}a^{-5}-11z^{5}a^{-7}+z^{5}a^{-9}-4a^{2}z^{4}+7z^{4}a^{-2}+41z^{4}a^{-4}+19z^{4}a^{-6}-6z^{4}a^{-8}-13z^{4}+az^{3}-6z^{3}a^{-1}-8z^{3}a^{-3}+8z^{3}a^{-5}+7z^{3}a^{-7}-2z^{3}a^{-9}+6a^{2}z^{2}-3z^{2}a^{-2}-22z^{2}a^{-4}-9z^{2}a^{-6}+z^{2}a^{-8}+15z^{2}+4az+6za^{-1}-4za^{-5}-2za^{-7}-4a^{2}-2a^{-2}+4a^{-4}+2a^{-6}-7-2az^{-1}-2a^{-1}z^{-1}+a^{2}z^{-2}+a^{-2}z^{-2}+2z^{-2}$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 17                       1 -1 15                     2   2 13                   4 1   -3 11                 6 2     4 9               7 4       -3 7             8 6         2 5           7 8           1 3         6 7             -1 1       4 9               5 -1     2 4                 -2 -3   1 5                   4 -5   1                     -1 -7 1                       1

Integral Khovanov Homology (db, data source)

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

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.

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