L11a393

L11a392

L11a394

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a393 at Knotilus!

Link Presentations

[edit Notes on L11a393's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {uvw^{5}-uvw^{4}+uvw^{3}-uvw^{2}+uvw-uv-uw^{5}+2uw^{4}-2uw^{3}+2uw^{2}-2uw+2u-2vw^{5}+2vw^{4}-2vw^{3}+2vw^{2}-2vw+v+w^{5}-w^{4}+w^{3}-w^{2}+w-1}{{\sqrt {u}}{\sqrt {v}}w^{5/2}}}$ (db)
Jones polynomial	$-q^{9}+3q^{8}-6q^{7}+7q^{6}-10q^{5}+11q^{4}-9q^{3}+9q^{2}+q^{-2}-5q-q^{-1}+5$ (db)
Signature	4 (db)
HOMFLY-PT polynomial	$z^{8}a^{-4}-2z^{6}a^{-2}+6z^{6}a^{-4}-z^{6}a^{-6}-11z^{4}a^{-2}+14z^{4}a^{-4}-4z^{4}a^{-6}+z^{4}-21z^{2}a^{-2}+20z^{2}a^{-4}-5z^{2}a^{-6}+5z^{2}-20a^{-2}+18a^{-4}-5a^{-6}+7-8a^{-2}z^{-2}+7a^{-4}z^{-2}-2a^{-6}z^{-2}+3z^{-2}$ (db)
Kauffman polynomial	$z^{10}a^{-2}+z^{10}a^{-4}+z^{9}a^{-1}+5z^{9}a^{-3}+4z^{9}a^{-5}-z^{8}a^{-2}+5z^{8}a^{-4}+7z^{8}a^{-6}+z^{8}-3z^{7}a^{-1}-18z^{7}a^{-3}-8z^{7}a^{-5}+7z^{7}a^{-7}-15z^{6}a^{-2}-34z^{6}a^{-4}-19z^{6}a^{-6}+7z^{6}a^{-8}-7z^{6}-5z^{5}a^{-1}+2z^{5}a^{-3}-9z^{5}a^{-5}-10z^{5}a^{-7}+6z^{5}a^{-9}+42z^{4}a^{-2}+51z^{4}a^{-4}+16z^{4}a^{-6}-8z^{4}a^{-8}+3z^{4}a^{-10}+18z^{4}+23z^{3}a^{-1}+44z^{3}a^{-3}+25z^{3}a^{-5}-3z^{3}a^{-7}-6z^{3}a^{-9}+z^{3}a^{-11}-47z^{2}a^{-2}-37z^{2}a^{-4}-12z^{2}a^{-6}-22z^{2}-24za^{-1}-45za^{-3}-21za^{-5}+3za^{-7}+3za^{-9}+28a^{-2}+22a^{-4}+7a^{-6}+a^{-8}+13+8a^{-1}z^{-1}+15a^{-3}z^{-1}+7a^{-5}z^{-1}-a^{-7}z^{-1}-a^{-9}z^{-1}-8a^{-2}z^{-2}-7a^{-4}z^{-2}-2a^{-6}z^{-2}-3z^{-2}$ (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 19                       1 -1 17                     2   2 15                   4 1   -3 13                 3 2     1 11               7 4       -3 9             4 3         1 7           5 7           2 5         4 4             0 3       4 8               4 1     1 1                 0 -1     4                   4 -3 1 1                     0 -5 1                       1

Integral Khovanov Homology (db, data source)

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

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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