L11a397

L11a396

L11a398

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a397 at Knotilus!

Link Presentations

[edit Notes on L11a397's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-3t(2)t(1)+3t(2)t(3)t(1)-3t(3)t(1)+4t(1)+3t(2)-4t(2)t(3)+3t(3)-3}{{\sqrt {t(1)}}{\sqrt {t(2)}}{\sqrt {t(3)}}}}$ (db)
Jones polynomial	$-q^{7}+3q^{6}-4q^{5}+5q^{4}-7q^{3}+8q^{2}-7q+7-4q^{-1}+4q^{-2}-q^{-3}+q^{-4}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-z^{2}a^{-6}+z^{4}a^{-4}+a^{4}z^{-2}+z^{2}a^{-4}+a^{4}+z^{4}a^{-2}-2a^{2}z^{2}-2a^{2}z^{-2}-3a^{2}+z^{4}+z^{2}+z^{-2}+2$ (db)
Kauffman polynomial	$z^{7}a^{-7}-4z^{5}a^{-7}+3z^{3}a^{-7}+3z^{8}a^{-6}-14z^{6}a^{-6}+16z^{4}a^{-6}-3z^{2}a^{-6}+3z^{9}a^{-5}-14z^{7}a^{-5}+18z^{5}a^{-5}-7z^{3}a^{-5}+z^{10}a^{-4}-z^{8}a^{-4}-8z^{6}a^{-4}+a^{4}z^{4}+10z^{4}a^{-4}-3a^{4}z^{2}-2z^{2}a^{-4}-a^{4}z^{-2}+3a^{4}+4z^{9}a^{-3}-17z^{7}a^{-3}+a^{3}z^{5}+22z^{5}a^{-3}-10z^{3}a^{-3}-3a^{3}z+2a^{3}z^{-1}+z^{10}a^{-2}-3z^{8}a^{-2}+a^{2}z^{6}+5z^{6}a^{-2}+2a^{2}z^{4}-6z^{4}a^{-2}-6a^{2}z^{2}+z^{2}a^{-2}-2a^{2}z^{-2}+5a^{2}+z^{9}a^{-1}+az^{7}-z^{7}a^{-1}+az^{5}-3az+2az^{-1}+z^{8}+z^{4}-3z^{2}-z^{-2}+3$ (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 15                       1 -1 13                     2   2 11                   2 1   -1 9                 3 2     1 7               4 2       -2 5             4 3         1 3           3 4           1 1         4 4             0 -1       3 6               3 -3     1 1                 0 -5     3                   3 -7 1 1                     0 -9 1                       1

Integral Khovanov Homology (db, data source)

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

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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