L11a513

L11a512

L11a514

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a513 at Knotilus!

Link Presentations

[edit Notes on L11a513's Link Presentations]

Planar diagram presentation	X8192 X16,5,17,6 X14,3,15,4 X4,15,5,16 X22,18,13,17 X10,20,11,19 X18,10,19,9 X20,12,21,11 X12,22,7,21 X2738 X6,13,1,14
Gauss code	{1, -10, 3, -4, 2, -11}, {10, -1, 7, -6, 8, -9}, {11, -3, 4, -2, 5, -7, 6, -8, 9, -5}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {u^{2}v^{2}w^{3}-u^{2}v^{2}w^{2}-u^{2}vw^{3}+2u^{2}vw^{2}-u^{2}vw-u^{2}w^{2}+2u^{2}w-2uv^{2}w^{3}+3uv^{2}w^{2}-uv^{2}w+uvw^{3}-4uvw^{2}+4uvw-uv+uw^{2}-3uw+2u-2v^{2}w^{2}+v^{2}w+vw^{2}-2vw+v+w-1}{uvw^{3/2}}}$ (db)
Jones polynomial	$q^{-6}-q^{5}-2q^{-5}+3q^{4}+5q^{-4}-5q^{3}-7q^{-3}+9q^{2}+11q^{-2}-11q-12q^{-1}+13$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{4}z^{4}+4a^{4}z^{2}+a^{4}z^{-2}+4a^{4}-2a^{2}z^{6}-z^{6}a^{-2}-10a^{2}z^{4}-4z^{4}a^{-2}-16a^{2}z^{2}-4z^{2}a^{-2}-2a^{2}z^{-2}-9a^{2}+z^{8}+6z^{6}+13z^{4}+12z^{2}+z^{-2}+5$ (db)
Kauffman polynomial	$a^{2}z^{10}+z^{10}+2a^{3}z^{9}+5az^{9}+3z^{9}a^{-1}+3a^{4}z^{8}+2a^{2}z^{8}+4z^{8}a^{-2}+3z^{8}+2a^{5}z^{7}-2a^{3}z^{7}-14az^{7}-6z^{7}a^{-1}+4z^{7}a^{-3}+a^{6}z^{6}-10a^{4}z^{6}-14a^{2}z^{6}-8z^{6}a^{-2}+3z^{6}a^{-4}-14z^{6}-6a^{5}z^{5}-8a^{3}z^{5}+14az^{5}+7z^{5}a^{-1}-8z^{5}a^{-3}+z^{5}a^{-5}-4a^{6}z^{4}+12a^{4}z^{4}+27a^{2}z^{4}+6z^{4}a^{-2}-7z^{4}a^{-4}+24z^{4}+3a^{5}z^{3}+14a^{3}z^{3}+4az^{3}-2z^{3}a^{-1}+3z^{3}a^{-3}-2z^{3}a^{-5}+4a^{6}z^{2}-11a^{4}z^{2}-25a^{2}z^{2}-z^{2}a^{-2}+3z^{2}a^{-4}-14z^{2}-9a^{3}z-9az-a^{6}+5a^{4}+11a^{2}-a^{-2}+5+2a^{3}z^{-1}+2az^{-1}-a^{4}z^{-2}-2a^{2}z^{-2}-z^{-2}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 11                       1 -1 9                     2   2 7                   3 1   -2 5                 6 2     4 3               6 4       -2 1             7 5         2 -1           7 8           1 -3         4 5             -1 -5       3 7               4 -7     2 4                 -2 -9     3                   3 -11 1 2                     -1 -13 1                       1

Integral Khovanov Homology (db, data source)

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

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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