L11a330

L11a329

L11a331

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a330 at Knotilus!

Link Presentations

[edit Notes on L11a330's Link Presentations]

Planar diagram presentation	X10,1,11,2 X12,4,13,3 X20,5,21,6 X6,9,7,10 X18,12,19,11 X16,8,17,7 X4,14,5,13 X22,16,9,15 X8,18,1,17 X2,19,3,20 X14,22,15,21
Gauss code	{1, -10, 2, -7, 3, -4, 6, -9}, {4, -1, 5, -2, 7, -11, 8, -6, 9, -5, 10, -3, 11, -8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(u-1)(v-1)\left(u^{2}v^{4}-u^{2}v^{3}+u^{2}v^{2}-uv^{4}+3uv^{3}-3uv^{2}+3uv-u+v^{2}-v+1\right)}{u^{3/2}v^{5/2}}}$ (db)
Jones polynomial	$19q^{9/2}-22q^{7/2}+21q^{5/2}-{\frac {1}{q^{5/2}}}-19q^{3/2}+{\frac {4}{q^{3/2}}}+q^{17/2}-4q^{15/2}+9q^{13/2}-14q^{11/2}+13{\sqrt {q}}-{\frac {9}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{7}a^{-5}+4z^{5}a^{-5}+5z^{3}a^{-5}+3za^{-5}+a^{-5}z^{-1}-z^{9}a^{-3}-6z^{7}a^{-3}-13z^{5}a^{-3}-13z^{3}a^{-3}-7za^{-3}-3a^{-3}z^{-1}+z^{7}a^{-1}+4z^{5}a^{-1}+5z^{3}a^{-1}+4za^{-1}+2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{4}a^{-10}+4z^{5}a^{-9}-z^{3}a^{-9}+9z^{6}a^{-8}-8z^{4}a^{-8}+3z^{2}a^{-8}+13z^{7}a^{-7}-17z^{5}a^{-7}+8z^{3}a^{-7}-za^{-7}+13z^{8}a^{-6}-18z^{6}a^{-6}+6z^{4}a^{-6}-3z^{2}a^{-6}+a^{-6}+9z^{9}a^{-5}-8z^{7}a^{-5}-9z^{5}a^{-5}+3z^{3}a^{-5}+2za^{-5}-a^{-5}z^{-1}+3z^{10}a^{-4}+12z^{8}a^{-4}-45z^{6}a^{-4}+36z^{4}a^{-4}-13z^{2}a^{-4}+3a^{-4}+15z^{9}a^{-3}-40z^{7}a^{-3}+28z^{5}a^{-3}-10z^{3}a^{-3}+7za^{-3}-3a^{-3}z^{-1}+3z^{10}a^{-2}+3z^{8}a^{-2}-31z^{6}a^{-2}+33z^{4}a^{-2}-10z^{2}a^{-2}+3a^{-2}+6z^{9}a^{-1}+az^{7}-18z^{7}a^{-1}-3az^{5}+13z^{5}a^{-1}+3az^{3}-z^{3}a^{-1}-az+3za^{-1}-2a^{-1}z^{-1}+4z^{8}-13z^{6}+12z^{4}-3z^{2}$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 18                       1 -1 16                     3   3 14                   6 1   -5 12                 8 3     5 10               11 6       -5 8             11 8         3 6           10 11           1 4         9 11             -2 2       6 12               6 0     3 7                 -4 -2   1 6                   5 -4   3                     -3 -6 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=2$ $i=4$ $r=-4$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-1$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=0$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{9}$ $r=1$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=2$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=3$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=4$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=5$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=6$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=7$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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