L11a327

L11a326

L11a328

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a327 at Knotilus!

Link Presentations

[edit Notes on L11a327's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {-u^{3}v^{5}+u^{3}v^{4}-u^{3}v^{3}+u^{3}v^{2}+u^{2}v^{5}-4u^{2}v^{4}+5u^{2}v^{3}-5u^{2}v^{2}+2u^{2}v+2uv^{4}-5uv^{3}+5uv^{2}-4uv+u+v^{3}-v^{2}+v-1}{u^{3/2}v^{5/2}}}$ (db)
Jones polynomial	$q^{9/2}-2q^{7/2}+5q^{5/2}-9q^{3/2}+11{\sqrt {q}}-{\frac {14}{\sqrt {q}}}+{\frac {13}{q^{3/2}}}-{\frac {12}{q^{5/2}}}+{\frac {9}{q^{7/2}}}-{\frac {5}{q^{9/2}}}+{\frac {2}{q^{11/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{3}z^{7}+6a^{3}z^{5}+13a^{3}z^{3}+10a^{3}z+2a^{3}z^{-1}-az^{9}-8az^{7}+z^{7}a^{-1}-25az^{5}+6z^{5}a^{-1}-36az^{3}+13z^{3}a^{-1}-22az+10za^{-1}-3az^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{7}z^{5}-3a^{7}z^{3}+2a^{7}z+2a^{6}z^{6}-4a^{6}z^{4}+a^{6}z^{2}+3a^{5}z^{7}-5a^{5}z^{5}+2a^{5}z^{3}-a^{5}z+4a^{4}z^{8}-10a^{4}z^{6}+z^{6}a^{-4}+14a^{4}z^{4}-4z^{4}a^{-4}-7a^{4}z^{2}+4z^{2}a^{-4}+3a^{3}z^{9}-6a^{3}z^{7}+2z^{7}a^{-3}+10a^{3}z^{5}-6z^{5}a^{-3}-10a^{3}z^{3}+4z^{3}a^{-3}+9a^{3}z-2a^{3}z^{-1}+a^{2}z^{10}+4a^{2}z^{8}+3z^{8}a^{-2}-17a^{2}z^{6}-9z^{6}a^{-2}+30a^{2}z^{4}+10z^{4}a^{-2}-18a^{2}z^{2}-7z^{2}a^{-2}+3a^{2}+a^{-2}+6az^{9}+3z^{9}a^{-1}-21az^{7}-10z^{7}a^{-1}+43az^{5}+21z^{5}a^{-1}-47az^{3}-28z^{3}a^{-1}+24az+12za^{-1}-3az^{-1}-a^{-1}z^{-1}+z^{10}+3z^{8}-15z^{6}+26z^{4}-21z^{2}+3$ (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 10                       1 -1 8                     1   1 6                   4 1   -3 4                 5 1     4 2               6 4       -2 0             8 5         3 -2           6 7           1 -4         6 7             -1 -6       3 6               3 -8     2 6                 -4 -10   1 4                   3 -12   1                     -1 -14 1                       1

Integral Khovanov Homology (db, data source)

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

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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