L11a44

L11a43

L11a45

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a44 at Knotilus!

Link Presentations

[edit Notes on L11a44's Link Presentations]

Planar diagram presentation	X6172 X10,4,11,3 X20,8,21,7 X22,15,5,16 X16,21,17,22 X14,10,15,9 X18,13,19,14 X12,19,13,20 X8,18,9,17 X2536 X4,12,1,11
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -9, 6, -2, 11, -8, 7, -6, 4, -5, 9, -7, 8, -3, 5, -4}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(1)-1)(t(2)-1)\left(t(2)^{4}-4t(2)^{3}+8t(2)^{2}-4t(2)+1\right)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$q^{11/2}-4q^{9/2}+10q^{7/2}-17q^{5/2}+20q^{3/2}-24{\sqrt {q}}+{\frac {23}{\sqrt {q}}}-{\frac {19}{q^{3/2}}}+{\frac {14}{q^{5/2}}}-{\frac {8}{q^{7/2}}}+{\frac {3}{q^{9/2}}}-{\frac {1}{q^{11/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-1}+3az^{5}-4z^{5}a^{-1}+z^{5}a^{-3}-3a^{3}z^{3}+8az^{3}-9z^{3}a^{-1}+2z^{3}a^{-3}+a^{5}z-5a^{3}z+9az-8za^{-1}+3za^{-3}+a^{5}z^{-1}-3a^{3}z^{-1}+4az^{-1}-2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-2a^{2}z^{10}-2z^{10}-4a^{3}z^{9}-13az^{9}-9z^{9}a^{-1}-3a^{4}z^{8}-9a^{2}z^{8}-17z^{8}a^{-2}-23z^{8}-a^{5}z^{7}+7a^{3}z^{7}+20az^{7}-5z^{7}a^{-1}-17z^{7}a^{-3}+10a^{4}z^{6}+41a^{2}z^{6}+26z^{6}a^{-2}-10z^{6}a^{-4}+67z^{6}+4a^{5}z^{5}+8a^{3}z^{5}+24az^{5}+50z^{5}a^{-1}+26z^{5}a^{-3}-4z^{5}a^{-5}-12a^{4}z^{4}-42a^{2}z^{4}-8z^{4}a^{-2}+7z^{4}a^{-4}-z^{4}a^{-6}-46z^{4}-6a^{5}z^{3}-24a^{3}z^{3}-49az^{3}-49z^{3}a^{-1}-18z^{3}a^{-3}+6a^{4}z^{2}+14a^{2}z^{2}-z^{2}a^{-2}-3z^{2}a^{-4}+10z^{2}+4a^{5}z+16a^{3}z+26az+21za^{-1}+7za^{-3}-a^{4}-3a^{2}-a^{-2}-2-a^{5}z^{-1}-3a^{3}z^{-1}-4az^{-1}-2a^{-1}z^{-1}$ (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                     3   3 8                   7 1   -6 6                 10 3     7 4               10 7       -3 2             14 10         4 0           11 12           1 -2         8 12             -4 -4       6 11               5 -6     2 8                 -6 -8   1 6                   5 -10   2                     -2 -12 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=0$ $i=2$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-3$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-2$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=-1$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=0$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{14}$ $r=1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=2$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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