L11a321

L11a320

L11a322

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a321 at Knotilus!

Link Presentations

[edit Notes on L11a321's Link Presentations]

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

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(t(1)-1)(t(2)-1)(t(1)t(2)-t(2)+2)(2t(2)t(1)-t(1)+1)}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$-{\frac {14}{q^{9/2}}}-q^{7/2}+{\frac {18}{q^{7/2}}}+3q^{5/2}-{\frac {21}{q^{5/2}}}-7q^{3/2}+{\frac {20}{q^{3/2}}}+{\frac {1}{q^{15/2}}}-{\frac {4}{q^{13/2}}}+{\frac {9}{q^{11/2}}}+12{\sqrt {q}}-{\frac {18}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-a^{5}z^{5}-2a^{5}z^{3}-a^{5}z+a^{3}z^{7}+3a^{3}z^{5}+2a^{3}z^{3}-a^{3}z+az^{7}+4az^{5}-z^{5}a^{-1}+7az^{3}-3z^{3}a^{-1}+5az-3za^{-1}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{8}z^{6}-2a^{8}z^{4}+a^{8}z^{2}+4a^{7}z^{7}-9a^{7}z^{5}+5a^{7}z^{3}-a^{7}z+7a^{6}z^{8}-16a^{6}z^{6}+9a^{6}z^{4}-3a^{6}z^{2}+6a^{5}z^{9}-8a^{5}z^{7}-5a^{5}z^{5}+5a^{5}z^{3}-a^{5}z+2a^{4}z^{10}+10a^{4}z^{8}-32a^{4}z^{6}+24a^{4}z^{4}-5a^{4}z^{2}+11a^{3}z^{9}-19a^{3}z^{7}+4a^{3}z^{5}+z^{5}a^{-3}+9a^{3}z^{3}-2z^{3}a^{-3}-4a^{3}z+za^{-3}+2a^{2}z^{10}+9a^{2}z^{8}-25a^{2}z^{6}+3z^{6}a^{-2}+21a^{2}z^{4}-5z^{4}a^{-2}-3a^{2}z^{2}+2z^{2}a^{-2}+5az^{9}-2az^{7}+5z^{7}a^{-1}-8az^{5}-7z^{5}a^{-1}+15az^{3}+4z^{3}a^{-1}-8az-3za^{-1}+az^{-1}+a^{-1}z^{-1}+6z^{8}-7z^{6}+3z^{4}-1$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 8                       1 1 6                     2   -2 4                   5 1   4 2                 7 2     -5 0               11 5       6 -2             11 9         -2 -4           10 9           1 -6         8 11             3 -8       6 10               -4 -10     3 8                 5 -12   1 6                   -5 -14   3                     3 -16 1                       -1

Integral Khovanov Homology (db, data source)

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

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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