L11a477

L11a476

L11a478

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a477 at Knotilus!

Link Presentations

[edit Notes on L11a477's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(u-1)(w-1)\left(v^{2}w^{3}-2v^{2}w^{2}+2vw^{2}-2vw+2w-1\right)}{{\sqrt {u}}vw^{2}}}$ (db)
Jones polynomial	$q^{-10}-2q^{-9}+5q^{-8}-8q^{-7}+11q^{-6}-12q^{-5}+13q^{-4}-10q^{-3}+9q^{-2}-q-5q^{-1}+3$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$a^{8}z^{4}+4a^{8}z^{2}+a^{8}z^{-2}+4a^{8}-2a^{6}z^{6}-10a^{6}z^{4}-17a^{6}z^{2}-2a^{6}z^{-2}-12a^{6}+a^{4}z^{8}+6a^{4}z^{6}+14a^{4}z^{4}+17a^{4}z^{2}+a^{4}z^{-2}+9a^{4}-a^{2}z^{6}-4a^{2}z^{4}-4a^{2}z^{2}-a^{2}$ (db)
Kauffman polynomial	$a^{12}z^{4}-2a^{12}z^{2}+a^{12}+2a^{11}z^{5}-2a^{11}z^{3}+3a^{10}z^{6}-2a^{10}z^{4}+4a^{9}z^{7}-5a^{9}z^{5}+4a^{9}z^{3}+4a^{8}z^{8}-6a^{8}z^{6}+6a^{8}z^{4}-6a^{8}z^{2}-a^{8}z^{-2}+5a^{8}+3a^{7}z^{9}-3a^{7}z^{7}-5a^{7}z^{5}+12a^{7}z^{3}-10a^{7}z+2a^{7}z^{-1}+a^{6}z^{10}+6a^{6}z^{8}-29a^{6}z^{6}+43a^{6}z^{4}-35a^{6}z^{2}-2a^{6}z^{-2}+15a^{6}+6a^{5}z^{9}-17a^{5}z^{7}+7a^{5}z^{5}+11a^{5}z^{3}-12a^{5}z+2a^{5}z^{-1}+a^{4}z^{10}+5a^{4}z^{8}-33a^{4}z^{6}+51a^{4}z^{4}-36a^{4}z^{2}-a^{4}z^{-2}+12a^{4}+3a^{3}z^{9}-9a^{3}z^{7}+a^{3}z^{5}+9a^{3}z^{3}-3a^{3}z+3a^{2}z^{8}-13a^{2}z^{6}+17a^{2}z^{4}-9a^{2}z^{2}+2a^{2}+az^{7}-4az^{5}+4az^{3}-az$ (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 3                       1 -1 1                     2   2 -1                   3 1   -2 -3                 6 2     4 -5               6 5       -1 -7             7 4         3 -9           5 6           1 -11         6 7             -1 -13       2 5               3 -15     3 6                 -3 -17   1 4                   3 -19   1                     -1 -21 1                       1

Integral Khovanov Homology (db, data source)

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

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.

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