L11n34

L11n33

L11n35

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n34 at Knotilus!

Link Presentations

[edit Notes on L11n34's Link Presentations]

Planar diagram presentation	X6172 X20,7,21,8 X4,21,1,22 X14,10,15,9 X8493 X5,13,6,12 X13,5,14,22 X18,16,19,15 X16,11,17,12 X10,17,11,18 X2,20,3,19
Gauss code	{1, -11, 5, -3}, {-6, -1, 2, -5, 4, -10, 9, 6, -7, -4, 8, -9, 10, -8, 11, -2, 3, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(1)-1)(t(2)-1)\left(t(2)^{4}-3t(2)^{3}+3t(2)^{2}-3t(2)+1\right)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-3q^{9/2}+{\frac {1}{q^{9/2}}}+7q^{7/2}-{\frac {4}{q^{7/2}}}-11q^{5/2}+{\frac {7}{q^{5/2}}}+14q^{3/2}-{\frac {12}{q^{3/2}}}-15{\sqrt {q}}+{\frac {14}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-za^{-5}-a^{-5}z^{-1}+z^{5}a^{-3}-a^{3}z^{3}+3z^{3}a^{-3}-a^{3}z+4za^{-3}+a^{3}z^{-1}+2a^{-3}z^{-1}-z^{7}a^{-1}+2az^{5}-4z^{5}a^{-1}+5az^{3}-6z^{3}a^{-1}+2az-4za^{-1}-az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$6z^{3}a^{-5}-5za^{-5}+a^{-5}z^{-1}+a^{4}z^{6}+3z^{6}a^{-4}-2a^{4}z^{4}+3z^{4}a^{-4}+a^{4}z^{2}-3z^{2}a^{-4}+a^{-4}+4a^{3}z^{7}+8z^{7}a^{-3}-11a^{3}z^{5}-16z^{5}a^{-3}+9a^{3}z^{3}+24z^{3}a^{-3}-a^{3}z-15za^{-3}-a^{3}z^{-1}+2a^{-3}z^{-1}+5a^{2}z^{8}+7z^{8}a^{-2}-10a^{2}z^{6}-10z^{6}a^{-2}+2a^{2}z^{4}+8z^{4}a^{-2}+a^{2}z^{2}-6z^{2}a^{-2}+a^{2}+3a^{-2}+2az^{9}+2z^{9}a^{-1}+9az^{7}+13z^{7}a^{-1}-35az^{5}-40z^{5}a^{-1}+27az^{3}+36z^{3}a^{-1}-5az-14za^{-1}-az^{-1}+a^{-1}z^{-1}+12z^{8}-24z^{6}+9z^{4}-3z^{2}+2$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 χ 10                   3 3 8                 4   -4 6               7 3   4 4             7 4     -3 2           8 7       1 0         8 9         1 -2       4 6           -2 -4     3 8             5 -6   1 4               -3 -8   3                 3 -10 1                   -1

Integral Khovanov Homology (db, data source)

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

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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