L11a474

L11a473

L11a475

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a474 at Knotilus!

Link Presentations

[edit Notes on L11a474's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(3)-1)\left(t(2)^{2}t(3)^{3}-t(2)t(3)^{3}-t(2)^{2}t(3)^{2}+t(2)t(3)^{2}-t(3)^{2}+t(2)^{2}t(3)-t(2)t(3)+t(3)+t(2)-1\right)}{{\sqrt {t(1)}}t(2)t(3)^{2}}}$ (db)
Jones polynomial	$-q^{9}+3q^{8}-6q^{7}+9q^{6}-11q^{5}+13q^{4}-11q^{3}+11q^{2}-7q+5-2q^{-1}+q^{-2}$ (db)
Signature	4 (db)
HOMFLY-PT polynomial	$-z^{6}a^{-6}-4z^{4}a^{-6}-5z^{2}a^{-6}-3a^{-6}+z^{8}a^{-4}+6z^{6}a^{-4}+14z^{4}a^{-4}+17z^{2}a^{-4}+a^{-4}z^{-2}+9a^{-4}-2z^{6}a^{-2}-10z^{4}a^{-2}-16z^{2}a^{-2}-2a^{-2}z^{-2}-10a^{-2}+z^{4}+4z^{2}+z^{-2}+4$ (db)
Kauffman polynomial	$z^{10}a^{-2}+z^{10}a^{-4}+2z^{9}a^{-1}+7z^{9}a^{-3}+5z^{9}a^{-5}+2z^{8}a^{-2}+10z^{8}a^{-4}+9z^{8}a^{-6}+z^{8}-10z^{7}a^{-1}-28z^{7}a^{-3}-8z^{7}a^{-5}+10z^{7}a^{-7}-29z^{6}a^{-2}-54z^{6}a^{-4}-22z^{6}a^{-6}+9z^{6}a^{-8}-6z^{6}+15z^{5}a^{-1}+27z^{5}a^{-3}-13z^{5}a^{-5}-19z^{5}a^{-7}+6z^{5}a^{-9}+61z^{4}a^{-2}+76z^{4}a^{-4}+12z^{4}a^{-6}-13z^{4}a^{-8}+3z^{4}a^{-10}+13z^{4}-5z^{3}a^{-1}+3z^{3}a^{-3}+19z^{3}a^{-5}+6z^{3}a^{-7}-4z^{3}a^{-9}+z^{3}a^{-11}-47z^{2}a^{-2}-44z^{2}a^{-4}-5z^{2}a^{-6}+5z^{2}a^{-8}-13z^{2}-4za^{-1}-9za^{-3}-6za^{-5}+za^{-9}+15a^{-2}+13a^{-4}+2a^{-6}-a^{-8}+6+2a^{-1}z^{-1}+2a^{-3}z^{-1}-2a^{-2}z^{-2}-a^{-4}z^{-2}-z^{-2}$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 19                       1 -1 17                     2   2 15                   4 1   -3 13                 5 2     3 11               6 4       -2 9             7 5         2 7           6 8           2 5         5 5             0 3       4 8               4 1     1 3                 -2 -1   1 4                   3 -3   1                     -1 -5 1                       1

Integral Khovanov Homology (db, data source)

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

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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