L11a503

L11a502

L11a504

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a503 at Knotilus!

Link Presentations

[edit Notes on L11a503's Link Presentations]

Planar diagram presentation	X8192 X2,9,3,10 X10,3,11,4 X14,5,15,6 X18,12,19,11 X12,18,7,17 X20,16,21,15 X22,20,13,19 X16,22,17,21 X6718 X4,13,5,14
Gauss code	{1, -2, 3, -11, 4, -10}, {10, -1, 2, -3, 5, -6}, {11, -4, 7, -9, 6, -5, 8, -7, 9, -8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(1)^{2}t(2)^{2}t(3)^{3}-t(1)t(3)^{3}-t(1)^{2}t(2)t(3)^{3}+t(1)t(2)t(3)^{3}+t(3)^{3}-3t(1)^{2}t(2)^{2}t(3)^{2}+2t(1)t(2)^{2}t(3)^{2}+2t(1)t(3)^{2}+2t(1)^{2}t(2)t(3)^{2}-4t(1)t(2)t(3)^{2}+2t(2)t(3)^{2}-3t(3)^{2}+3t(1)^{2}t(2)^{2}t(3)-2t(1)t(2)^{2}t(3)-2t(1)t(3)-2t(1)^{2}t(2)t(3)+4t(1)t(2)t(3)-2t(2)t(3)+3t(3)-t(1)^{2}t(2)^{2}+t(1)t(2)^{2}-t(1)t(2)+t(2)-1}{t(1)t(2)t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{4}+2q^{3}-5q^{2}+9q-11+15q^{-1}-14q^{-2}+14q^{-3}-10q^{-4}+7q^{-5}-3q^{-6}+q^{-7}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^{4}z^{6}+4a^{4}z^{4}+6a^{4}z^{2}+2a^{4}z^{-2}+5a^{4}-a^{2}z^{8}-6a^{2}z^{6}-15a^{2}z^{4}-z^{4}a^{-2}-21a^{2}z^{2}-4z^{2}a^{-2}-5a^{2}z^{-2}-a^{-2}z^{-2}-16a^{2}-4a^{-2}+2z^{6}+10z^{4}+18z^{2}+4z^{-2}+15$ (db)
Kauffman polynomial	$a^{2}z^{10}+z^{10}+4a^{3}z^{9}+6az^{9}+2z^{9}a^{-1}+7a^{4}z^{8}+9a^{2}z^{8}+2z^{8}a^{-2}+4z^{8}+7a^{5}z^{7}-a^{3}z^{7}-12az^{7}-3z^{7}a^{-1}+z^{7}a^{-3}+6a^{6}z^{6}-12a^{4}z^{6}-35a^{2}z^{6}-8z^{6}a^{-2}-25z^{6}+3a^{7}z^{5}-8a^{5}z^{5}-15a^{3}z^{5}-11az^{5}-12z^{5}a^{-1}-5z^{5}a^{-3}+a^{8}z^{4}-8a^{6}z^{4}+11a^{4}z^{4}+47a^{2}z^{4}+10z^{4}a^{-2}+37z^{4}-2a^{7}z^{3}+2a^{5}z^{3}+23a^{3}z^{3}+39az^{3}+28z^{3}a^{-1}+8z^{3}a^{-3}-a^{8}z^{2}+6a^{6}z^{2}-9a^{4}z^{2}-39a^{2}z^{2}-7z^{2}a^{-2}-30z^{2}+a^{5}z-16a^{3}z-33az-21za^{-1}-5za^{-3}-2a^{6}+6a^{4}+20a^{2}+4a^{-2}+17+5a^{3}z^{-1}+9az^{-1}+5a^{-1}z^{-1}+a^{-3}z^{-1}-2a^{4}z^{-2}-5a^{2}z^{-2}-a^{-2}z^{-2}-4z^{-2}$ (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 9                       1 -1 7                     1   1 5                   4 1   -3 3                 5 1     4 1               6 4       -2 -1             9 5         4 -3           7 8           1 -5         7 7             0 -7       4 8               4 -9     3 6                 -3 -11     4                   4 -13 1 3                     -2 -15 1                       1

Integral Khovanov Homology (db, data source)

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

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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