L11a35

L11a34

L11a36

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a35 at Knotilus!

Link Presentations

[edit Notes on L11a35's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(2)^{5}+2t(1)t(2)^{4}-6t(2)^{4}-7t(1)t(2)^{3}+9t(2)^{3}+9t(1)t(2)^{2}-7t(2)^{2}-6t(1)t(2)+2t(2)+t(1)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-3q^{9/2}+{\frac {3}{q^{9/2}}}+6q^{7/2}-{\frac {7}{q^{7/2}}}-10q^{5/2}+{\frac {10}{q^{5/2}}}+14q^{3/2}-{\frac {14}{q^{3/2}}}+q^{11/2}-{\frac {1}{q^{11/2}}}-16{\sqrt {q}}+{\frac {15}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$a^{5}z+a^{5}z^{-1}+za^{-5}-3a^{3}z^{3}-2z^{3}a^{-3}-5a^{3}z-2a^{3}z^{-1}-za^{-3}+a^{-3}z^{-1}+2az^{5}+z^{5}a^{-1}+5az^{3}-z^{3}a^{-1}+6az+3az^{-1}-4za^{-1}-3a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{4}a^{-6}-z^{2}a^{-6}+a^{5}z^{7}-4a^{5}z^{5}+3z^{5}a^{-5}+6a^{5}z^{3}-3z^{3}a^{-5}-4a^{5}z+za^{-5}+a^{5}z^{-1}+3a^{4}z^{8}-11a^{4}z^{6}+5z^{6}a^{-4}+12a^{4}z^{4}-5z^{4}a^{-4}-4a^{4}z^{2}+3z^{2}a^{-4}-a^{-4}+3a^{3}z^{9}-5a^{3}z^{7}+6z^{7}a^{-3}-11a^{3}z^{5}-5z^{5}a^{-3}+22a^{3}z^{3}+z^{3}a^{-3}-11a^{3}z-za^{-3}+2a^{3}z^{-1}+a^{-3}z^{-1}+a^{2}z^{10}+8a^{2}z^{8}+6z^{8}a^{-2}-36a^{2}z^{6}-5z^{6}a^{-2}+38a^{2}z^{4}-5z^{4}a^{-2}-14a^{2}z^{2}+8z^{2}a^{-2}+2a^{2}-2a^{-2}+7az^{9}+4z^{9}a^{-1}-11az^{7}+z^{7}a^{-1}-18az^{5}-19z^{5}a^{-1}+35az^{3}+23z^{3}a^{-1}-18az-13za^{-1}+3az^{-1}+3a^{-1}z^{-1}+z^{10}+11z^{8}-35z^{6}+27z^{4}-6z^{2}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 12                       1 -1 10                     2   2 8                   4 1   -3 6                 6 2     4 4               8 4       -4 2             8 6         2 0           8 9           1 -2         6 7             -1 -4       4 8               4 -6     3 6                 -3 -8   1 5                   4 -10   2                     -2 -12 1                       1

Integral Khovanov Homology (db, data source)

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

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.

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