L11a358

L11a357

L11a359

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a358 at Knotilus!

Link Presentations

[edit Notes on L11a358's Link Presentations]

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(2)^{3}t(1)^{4}-t(2)^{2}t(1)^{4}+t(2)^{4}t(1)^{3}-3t(2)^{3}t(1)^{3}+3t(2)^{2}t(1)^{3}-t(2)t(1)^{3}-t(2)^{4}t(1)^{2}+3t(2)^{3}t(1)^{2}-3t(2)^{2}t(1)^{2}+3t(2)t(1)^{2}-t(1)^{2}-t(2)^{3}t(1)+3t(2)^{2}t(1)-3t(2)t(1)+t(1)-t(2)^{2}+t(2)}{t(1)^{2}t(2)^{2}}}$ (db)
Jones polynomial	$-q^{15/2}+3q^{13/2}-5q^{11/2}+7q^{9/2}-9q^{7/2}+9q^{5/2}-9q^{3/2}+7{\sqrt {q}}-{\frac {6}{\sqrt {q}}}+{\frac {3}{q^{3/2}}}-{\frac {2}{q^{5/2}}}+{\frac {1}{q^{7/2}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{7}a^{-1}+z^{7}a^{-3}-az^{5}+5z^{5}a^{-1}+4z^{5}a^{-3}-z^{5}a^{-5}-4az^{3}+8z^{3}a^{-1}+3z^{3}a^{-3}-3z^{3}a^{-5}-3az+6za^{-1}-za^{-3}-za^{-5}+a^{-1}z^{-1}-a^{-3}z^{-1}$ (db)
Kauffman polynomial	$-z^{10}a^{-2}-z^{10}-2az^{9}-5z^{9}a^{-1}-3z^{9}a^{-3}-a^{2}z^{8}-2z^{8}a^{-2}-5z^{8}a^{-4}+2z^{8}+12az^{7}+24z^{7}a^{-1}+6z^{7}a^{-3}-6z^{7}a^{-5}+6a^{2}z^{6}+19z^{6}a^{-2}+11z^{6}a^{-4}-6z^{6}a^{-6}+8z^{6}-24az^{5}-37z^{5}a^{-1}+2z^{5}a^{-3}+10z^{5}a^{-5}-5z^{5}a^{-7}-11a^{2}z^{4}-23z^{4}a^{-2}-4z^{4}a^{-4}+7z^{4}a^{-6}-3z^{4}a^{-8}-20z^{4}+19az^{3}+25z^{3}a^{-1}-z^{3}a^{-3}-2z^{3}a^{-5}+4z^{3}a^{-7}-z^{3}a^{-9}+6a^{2}z^{2}+7z^{2}a^{-2}+z^{2}a^{-4}-z^{2}a^{-6}+z^{2}a^{-8}+10z^{2}-5az-9za^{-1}-3za^{-3}-za^{-7}-a^{-2}+a^{-1}z^{-1}+a^{-3}z^{-1}$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 16                       1 1 14                     2   -2 12                   3 1   2 10                 4 2     -2 8               5 3       2 6             5 5         0 4           4 4           0 2         4 6             2 0       2 3               -1 -2     1 4                 3 -4   1 2                   -1 -6   1                     1 -8 1                       -1

Integral Khovanov Homology (db, data source)

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

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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