L11a408

L11a407

L11a409

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a408 at Knotilus!

Link Presentations

[edit Notes on L11a408's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(1)t(3)^{2}t(2)^{2}-2t(3)^{2}t(2)^{2}+2t(1)t(2)^{2}-3t(1)t(3)t(2)^{2}+5t(3)t(2)^{2}-2t(2)^{2}-3t(1)t(3)^{2}t(2)+5t(3)^{2}t(2)-5t(1)t(2)+8t(1)t(3)t(2)-8t(3)t(2)+3t(2)+2t(1)t(3)^{2}-2t(3)^{2}+2t(1)-5t(1)t(3)+3t(3)-1}{{\sqrt {t(1)}}t(2)t(3)}}$ (db)
Jones polynomial	$-q^{5}+4q^{4}-8q^{3}+14q^{2}-17q+21-19q^{-1}+17q^{-2}-12q^{-3}+7q^{-4}-3q^{-5}+q^{-6}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{6}-3z^{2}a^{4}-2a^{4}+3z^{4}a^{2}+4z^{2}a^{2}+a^{2}z^{-2}+3a^{2}-z^{6}-2z^{4}-5z^{2}-2z^{-2}-4+2z^{4}a^{-2}+2z^{2}a^{-2}+a^{-2}z^{-2}+2a^{-2}-z^{2}a^{-4}$ (db)
Kauffman polynomial	$a^{6}z^{6}-3a^{6}z^{4}+3a^{6}z^{2}-a^{6}+3a^{5}z^{7}-8a^{5}z^{5}+z^{5}a^{-5}+7a^{5}z^{3}-z^{3}a^{-5}-2a^{5}z+4a^{4}z^{8}-6a^{4}z^{6}+4z^{6}a^{-4}-2a^{4}z^{4}-6z^{4}a^{-4}+5a^{4}z^{2}+3z^{2}a^{-4}-a^{4}+3a^{3}z^{9}+3a^{3}z^{7}+7z^{7}a^{-3}-19a^{3}z^{5}-10z^{5}a^{-3}+18a^{3}z^{3}+4z^{3}a^{-3}-6a^{3}z+a^{2}z^{10}+10a^{2}z^{8}+7z^{8}a^{-2}-20a^{2}z^{6}-5z^{6}a^{-2}+5a^{2}z^{4}-5z^{4}a^{-2}+6a^{2}z^{2}+6z^{2}a^{-2}+a^{2}z^{-2}+a^{-2}z^{-2}-3a^{2}-4a^{-2}+7az^{9}+4z^{9}a^{-1}+7z^{7}a^{-1}-20az^{5}-20z^{5}a^{-1}+14az^{3}+8z^{3}a^{-1}+4za^{-1}-2az^{-1}-2a^{-1}z^{-1}+z^{10}+13z^{8}-22z^{6}+5z^{4}+7z^{2}+2z^{-2}-6$ (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 11                       1 -1 9                     3   3 7                   5 1   -4 5                 9 3     6 3               10 7       -3 1             11 7         4 -1           9 11           2 -3         8 10             -2 -5       4 9               5 -7     3 8                 -5 -9   1 5                   4 -11   2                     -2 -13 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{3}$ $r=-3$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-2$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=-1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=0$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{11}$ $r=1$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=2$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{9}$ $r=3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

L11a407 L11a409