L11a59

L11a58

L11a60

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a59 at Knotilus!

Link Presentations

[edit Notes on L11a59's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(1)t(2)^{5}-2t(2)^{5}-4t(1)t(2)^{4}+6t(2)^{4}+7t(1)t(2)^{3}-7t(2)^{3}-7t(1)t(2)^{2}+7t(2)^{2}+6t(1)t(2)-4t(2)-2t(1)+1}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$q^{11/2}-4q^{9/2}+8q^{7/2}-13q^{5/2}+16q^{3/2}-17{\sqrt {q}}+{\frac {17}{\sqrt {q}}}-{\frac {14}{q^{3/2}}}+{\frac {9}{q^{5/2}}}-{\frac {6}{q^{7/2}}}+{\frac {2}{q^{9/2}}}-{\frac {1}{q^{11/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$a^{5}z+2a^{5}z^{-1}+z^{5}a^{-3}-3a^{3}z^{3}+2z^{3}a^{-3}-8a^{3}z+za^{-3}-4a^{3}z^{-1}-z^{7}a^{-1}+3az^{5}-4z^{5}a^{-1}+10az^{3}-7z^{3}a^{-1}+10az-6za^{-1}+3az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-a^{2}z^{10}-z^{10}-2a^{3}z^{9}-7az^{9}-5z^{9}a^{-1}-2a^{4}z^{8}-5a^{2}z^{8}-10z^{8}a^{-2}-13z^{8}-a^{5}z^{7}+2a^{3}z^{7}+12az^{7}-2z^{7}a^{-1}-11z^{7}a^{-3}+7a^{4}z^{6}+22a^{2}z^{6}+15z^{6}a^{-2}-8z^{6}a^{-4}+38z^{6}+5a^{5}z^{5}+12a^{3}z^{5}+9az^{5}+21z^{5}a^{-1}+15z^{5}a^{-3}-4z^{5}a^{-5}-6a^{4}z^{4}-19a^{2}z^{4}-7z^{4}a^{-2}+7z^{4}a^{-4}-z^{4}a^{-6}-28z^{4}-9a^{5}z^{3}-23a^{3}z^{3}-22az^{3}-16z^{3}a^{-1}-6z^{3}a^{-3}+2z^{3}a^{-5}-a^{4}z^{2}+a^{2}z^{2}+z^{2}a^{-2}-z^{2}a^{-4}+4z^{2}+7a^{5}z+15a^{3}z+12az+4za^{-1}+2a^{4}+3a^{2}+a^{-2}+3-2a^{5}z^{-1}-4a^{3}z^{-1}-3az^{-1}-a^{-1}z^{-1}$ (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                   5 1   -4 6                 8 3     5 4               8 5       -3 2             9 8         1 0           9 9           0 -2         5 8             -3 -4       4 9               5 -6     2 5                 -3 -8   1 5                   4 -10   1                     -1 -12 1                       1

Integral Khovanov Homology (db, data source)

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

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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