L11n400

L11n399

L11n401

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n400 at Knotilus!

Link Presentations

[edit Notes on L11n400's Link Presentations]

Planar diagram presentation	X6172 X10,3,11,4 X15,22,16,19 X7,20,8,21 X19,8,20,9 X18,14,5,13 X14,12,15,11 X12,18,13,17 X21,16,22,17 X2536 X4,9,1,10
Gauss code	{1, -10, 2, -11}, {-5, 4, -9, 3}, {10, -1, -4, 5, 11, -2, 7, -8, 6, -7, -3, 9, 8, -6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(3)-1)\left(t(3)^{4}+t(1)t(2)t(3)^{3}-t(2)t(3)^{3}+t(3)^{3}+t(1)t(3)^{2}-t(1)t(2)t(3)^{2}+t(2)t(3)^{2}-t(3)^{2}-t(1)t(3)+t(1)t(2)t(3)+t(3)+t(1)t(2)\right)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{5/2}}}$ (db)
Jones polynomial	$-q^{2}+3q-5+6q^{-1}-6q^{-2}+6q^{-3}-4q^{-4}+3q^{-5}+q^{-6}+q^{-8}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$2a^{8}z^{-2}+a^{8}-2z^{2}a^{6}-5a^{6}z^{-2}-7a^{6}+3z^{2}a^{4}+4a^{4}z^{-2}+6a^{4}+z^{6}a^{2}+4z^{4}a^{2}+5z^{2}a^{2}-a^{2}z^{-2}+a^{2}-z^{4}-2z^{2}-1$ (db)
Kauffman polynomial	$z^{8}a^{8}-8z^{6}a^{8}+21z^{4}a^{8}-24z^{2}a^{8}-2a^{8}z^{-2}+12a^{8}+z^{7}a^{7}-10z^{5}a^{7}+25z^{3}a^{7}-21za^{7}+5a^{7}z^{-1}+z^{8}a^{6}-11z^{6}a^{6}+35z^{4}a^{6}-41z^{2}a^{6}-5a^{6}z^{-2}+23a^{6}+2z^{7}a^{5}-15z^{5}a^{5}+42z^{3}a^{5}-35za^{5}+9a^{5}z^{-1}+z^{8}a^{4}-3z^{6}a^{4}+9z^{4}a^{4}-12z^{2}a^{4}-4a^{4}z^{-2}+12a^{4}+4z^{7}a^{3}-11z^{5}a^{3}+17z^{3}a^{3}-15za^{3}+5a^{3}z^{-1}+z^{8}a^{2}+3z^{6}a^{2}-12z^{4}a^{2}+8z^{2}a^{2}-a^{2}z^{-2}-a^{2}+3z^{7}a-5z^{5}a-2z^{3}a+az^{-1}+3z^{6}-7z^{4}+3z^{2}-1+z^{5}a^{-1}-2z^{3}a^{-1}+za^{-1}$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 5                       1 -1 3                     2   2 1                   3 1   -2 -1                 3 2     1 -3               4 4       0 -5           1 3 2         0 -7           2 4           2 -9       1 3 3             -1 -11         4               4 -13     1                   1 -15 1                       1 -17 1                       1

Integral Khovanov Homology (db, data source)

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

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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