L11a328

L11a327

L11a329

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a328 at Knotilus!

Link Presentations

[edit Notes on L11a328's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(1)-1)(t(2)-1)\left(2t(2)^{2}t(1)^{2}-t(2)t(1)^{2}-2t(2)^{2}t(1)+4t(2)t(1)-2t(1)-t(2)+2\right)}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$-3q^{9/2}+{\frac {4}{q^{9/2}}}+6q^{7/2}-{\frac {8}{q^{7/2}}}-11q^{5/2}+{\frac {12}{q^{5/2}}}+15q^{3/2}-{\frac {16}{q^{3/2}}}+q^{11/2}-{\frac {1}{q^{11/2}}}-18{\sqrt {q}}+{\frac {17}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$a^{3}z^{5}+z^{5}a^{-3}+2a^{3}z^{3}+3z^{3}a^{-3}+2za^{-3}-az^{7}-z^{7}a^{-1}-3az^{5}-4z^{5}a^{-1}-az^{3}-6z^{3}a^{-1}+2az-4za^{-1}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{4}a^{-6}-z^{2}a^{-6}+a^{5}z^{7}-3a^{5}z^{5}+3z^{5}a^{-5}+3a^{5}z^{3}-3z^{3}a^{-5}-a^{5}z+za^{-5}+4a^{4}z^{8}-14a^{4}z^{6}+5z^{6}a^{-4}+14a^{4}z^{4}-4z^{4}a^{-4}-4a^{4}z^{2}+z^{2}a^{-4}+5a^{3}z^{9}-14a^{3}z^{7}+7z^{7}a^{-3}+5a^{3}z^{5}-8z^{5}a^{-3}+6a^{3}z^{3}+4z^{3}a^{-3}-3a^{3}z-za^{-3}+2a^{2}z^{10}+6a^{2}z^{8}+8z^{8}a^{-2}-37a^{2}z^{6}-13z^{6}a^{-2}+38a^{2}z^{4}+8z^{4}a^{-2}-10a^{2}z^{2}-z^{2}a^{-2}+11az^{9}+6z^{9}a^{-1}-28az^{7}-6z^{7}a^{-1}+9az^{5}-10z^{5}a^{-1}+11az^{3}+15z^{3}a^{-1}-6az-6za^{-1}+az^{-1}+a^{-1}z^{-1}+2z^{10}+10z^{8}-41z^{6}+37z^{4}-9z^{2}-1$ (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                 7 2     5 4               8 4       -4 2             10 7         3 0           9 10           1 -2         7 8             -1 -4       5 9               4 -6     3 7                 -4 -8   1 5                   4 -10   3                     -3 -12 1                       1

Integral Khovanov Homology (db, data source)

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

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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