L11a500

L11a499

L11a501

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a500 at Knotilus!

Link Presentations

[edit Notes on L11a500's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(w-1)\left(2uvw^{3}-4uvw^{2}+3uvw-uv-2uw^{3}+4uw^{2}-4uw-4vw^{3}+4vw^{2}-2vw-w^{4}+3w^{3}-4w^{2}+2w\right)}{{\sqrt {u}}{\sqrt {v}}w^{5/2}}}$ (db)
Jones polynomial	$-q^{8}+5q^{7}-12q^{6}+18q^{5}-23q^{4}+27q^{3}-25q^{2}+22q-14+9q^{-1}-3q^{-2}+q^{-3}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{6}a^{-2}+z^{6}a^{-4}-z^{4}a^{-2}+z^{4}a^{-4}-z^{4}a^{-6}-2z^{4}+a^{2}z^{2}-7z^{2}a^{-2}+3z^{2}a^{-4}-z^{2}+a^{2}-10a^{-2}+7a^{-4}-a^{-6}+3-5a^{-2}z^{-2}+4a^{-4}z^{-2}-a^{-6}z^{-2}+2z^{-2}$ (db)
Kauffman polynomial	$z^{5}a^{-9}+5z^{6}a^{-8}-3z^{4}a^{-8}+12z^{7}a^{-7}-16z^{5}a^{-7}+5z^{3}a^{-7}-2za^{-7}+a^{-7}z^{-1}+15z^{8}a^{-6}-23z^{6}a^{-6}+11z^{4}a^{-6}-4z^{2}a^{-6}-a^{-6}z^{-2}+2a^{-6}+9z^{9}a^{-5}+3z^{7}a^{-5}-30z^{5}a^{-5}+26z^{3}a^{-5}-15za^{-5}+5a^{-5}z^{-1}+2z^{10}a^{-4}+25z^{8}a^{-4}-60z^{6}a^{-4}+47z^{4}a^{-4}-25z^{2}a^{-4}-4a^{-4}z^{-2}+14a^{-4}+14z^{9}a^{-3}-11z^{7}a^{-3}-31z^{5}a^{-3}+52z^{3}a^{-3}-33za^{-3}+9a^{-3}z^{-1}+2z^{10}a^{-2}+16z^{8}a^{-2}+a^{2}z^{6}-47z^{6}a^{-2}-3a^{2}z^{4}+53z^{4}a^{-2}+3a^{2}z^{2}-40z^{2}a^{-2}-5a^{-2}z^{-2}-a^{2}+21a^{-2}+5z^{9}a^{-1}+3az^{7}+z^{7}a^{-1}-6az^{5}-24z^{5}a^{-1}+3az^{3}+34z^{3}a^{-1}-20za^{-1}+5a^{-1}z^{-1}+6z^{8}-14z^{6}+17z^{4}-16z^{2}-2z^{-2}+9$ (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                     4   4 13                   8 1   -7 11                 10 4     6 9               14 9       -5 7             13 9         4 5           12 14           2 3         10 13             -3 1       6 14               8 -1     3 8                 -5 -3     6                   6 -5 1 3                     -2 -7 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=1$ $i=3$ $r=-4$ ${\mathbb {Z} }$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{3}$ $r=-2$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-1$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=0$ ${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{10}$ $r=1$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=2$ ${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{13}$ $r=3$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{14}$ ${\mathbb {Z} }^{14}$ $r=4$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{10}$ $r=5$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=6$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=7$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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