L11a419

L11a418

L11a420

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a419 at Knotilus!

Link Presentations

[edit Notes on L11a419's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {2t(1)t(3)^{2}t(2)^{2}-2t(3)^{2}t(2)^{2}+t(1)t(2)^{2}-3t(1)t(3)t(2)^{2}+4t(3)t(2)^{2}-2t(2)^{2}-4t(1)t(3)^{2}t(2)+3t(3)^{2}t(2)-3t(1)t(2)+6t(1)t(3)t(2)-6t(3)t(2)+4t(2)+2t(1)t(3)^{2}-t(3)^{2}+2t(1)-4t(1)t(3)+3t(3)-2}{{\sqrt {t(1)}}t(2)t(3)}}$ (db)
Jones polynomial	$-q^{8}+4q^{7}-8q^{6}+12q^{5}-16q^{4}+18q^{3}+q^{-3}-16q^{2}-2q^{-2}+15q+6q^{-1}-9$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-z^{4}a^{-6}-z^{2}a^{-6}+z^{6}a^{-4}+2z^{4}a^{-4}+z^{2}a^{-4}-a^{-4}+z^{6}a^{-2}+2z^{4}a^{-2}+a^{2}z^{2}+3z^{2}a^{-2}+a^{2}z^{-2}+a^{-2}z^{-2}+2a^{2}+4a^{-2}-2z^{4}-5z^{2}-2z^{-2}-5$ (db)
Kauffman polynomial	$z^{10}a^{-2}+z^{10}a^{-4}+2z^{9}a^{-1}+6z^{9}a^{-3}+4z^{9}a^{-5}+4z^{8}a^{-2}+9z^{8}a^{-4}+7z^{8}a^{-6}+2z^{8}+2az^{7}+z^{7}a^{-1}-7z^{7}a^{-3}+z^{7}a^{-5}+7z^{7}a^{-7}+a^{2}z^{6}-4z^{6}a^{-2}-18z^{6}a^{-4}-10z^{6}a^{-6}+4z^{6}a^{-8}+z^{6}-5az^{5}-6z^{5}a^{-1}+3z^{5}a^{-3}-9z^{5}a^{-5}-12z^{5}a^{-7}+z^{5}a^{-9}-4a^{2}z^{4}-10z^{4}a^{-2}+11z^{4}a^{-4}+4z^{4}a^{-6}-6z^{4}a^{-8}-15z^{4}+2az^{3}-z^{3}a^{-1}-4z^{3}a^{-3}+5z^{3}a^{-5}+5z^{3}a^{-7}-z^{3}a^{-9}+6a^{2}z^{2}+15z^{2}a^{-2}-z^{2}a^{-6}+z^{2}a^{-8}+19z^{2}+3az+5za^{-1}+3za^{-3}+za^{-5}-4a^{2}-8a^{-2}-2a^{-4}-9-2az^{-1}-2a^{-1}z^{-1}+a^{2}z^{-2}+a^{-2}z^{-2}+2z^{-2}$ (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 17                       1 -1 15                     3   3 13                   5 1   -4 11                 7 3     4 9               9 5       -4 7             9 7         2 5           8 10           2 3         7 8             -1 1       4 10               6 -1     2 5                 -3 -3   1 5                   4 -5   1                     -1 -7 1                       1

Integral Khovanov Homology (db, data source)

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

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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