L11a331

L11a330

L11a332

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a331 at Knotilus!

Link Presentations

[edit Notes on L11a331's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(1)^{3}t(2)^{5}-3t(1)^{3}t(2)^{4}+3t(1)^{2}t(2)^{4}+3t(1)^{3}t(2)^{3}-6t(1)^{2}t(2)^{3}+4t(1)t(2)^{3}+4t(1)^{2}t(2)^{2}-6t(1)t(2)^{2}+3t(2)^{2}+3t(1)t(2)-3t(2)+1}{t(1)^{3/2}t(2)^{5/2}}}$ (db)
Jones polynomial	${\sqrt {q}}-{\frac {3}{\sqrt {q}}}+{\frac {5}{q^{3/2}}}-{\frac {8}{q^{5/2}}}+{\frac {10}{q^{7/2}}}-{\frac {12}{q^{9/2}}}+{\frac {12}{q^{11/2}}}-{\frac {11}{q^{13/2}}}+{\frac {8}{q^{15/2}}}-{\frac {6}{q^{17/2}}}+{\frac {3}{q^{19/2}}}-{\frac {1}{q^{21/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^{7}z^{7}+5a^{7}z^{5}+8a^{7}z^{3}+6a^{7}z+2a^{7}z^{-1}-a^{5}z^{9}-7a^{5}z^{7}-18a^{5}z^{5}-22a^{5}z^{3}-13a^{5}z-3a^{5}z^{-1}+a^{3}z^{7}+5a^{3}z^{5}+7a^{3}z^{3}+3a^{3}z+a^{3}z^{-1}$ (db)
Kauffman polynomial	$-z^{3}a^{13}-3z^{4}a^{12}-6z^{5}a^{11}+5z^{3}a^{11}-2za^{11}-8z^{6}a^{10}+10z^{4}a^{10}-2z^{2}a^{10}-9z^{7}a^{9}+17z^{5}a^{9}-7z^{3}a^{9}+3za^{9}-8z^{8}a^{8}+18z^{6}a^{8}-6z^{4}a^{8}+z^{2}a^{8}-6z^{9}a^{7}+17z^{7}a^{7}-12z^{5}a^{7}+10z^{3}a^{7}-8za^{7}+2a^{7}z^{-1}-2z^{10}a^{6}-2z^{8}a^{6}+31z^{6}a^{6}-43z^{4}a^{6}+18z^{2}a^{6}-3a^{6}-9z^{9}a^{5}+42z^{7}a^{5}-62z^{5}a^{5}+39z^{3}a^{5}-16za^{5}+3a^{5}z^{-1}-2z^{10}a^{4}+5z^{8}a^{4}+10z^{6}a^{4}-32z^{4}a^{4}+20z^{2}a^{4}-3a^{4}-3z^{9}a^{3}+16z^{7}a^{3}-27z^{5}a^{3}+16z^{3}a^{3}-3za^{3}+a^{3}z^{-1}-z^{8}a^{2}+5z^{6}a^{2}-8z^{4}a^{2}+5z^{2}a^{2}-a^{2}$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 2                       1 -1 0                     2   2 -2                   3 1   -2 -4                 5 2     3 -6               6 4       -2 -8             6 4         2 -10           6 6           0 -12         5 6             -1 -14       3 6               3 -16     3 5                 -2 -18     3                   3 -20 1 3                     -2 -22 1                       1

Integral Khovanov Homology (db, data source)

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

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.

L11a330 L11a332