L11a495

L11a494

L11a496

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a495 at Knotilus!

Link Presentations

[edit Notes on L11a495's Link Presentations]

Planar diagram presentation	X6172 X16,7,17,8 X4,17,1,18 X22,12,19,11 X10,4,11,3 X20,5,21,6 X18,21,5,22 X12,20,13,19 X14,9,15,10 X2,14,3,13 X8,15,9,16
Gauss code	{1, -10, 5, -3}, {8, -6, 7, -4}, {6, -1, 2, -11, 9, -5, 4, -8, 10, -9, 11, -2, 3, -7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2(u-1)(v-1)(w-1)^{3}}{{\sqrt {u}}{\sqrt {v}}w^{3/2}}}$ (db)
Jones polynomial	$-q^{-7}+3q^{-6}-7q^{-5}+q^{4}+12q^{-4}-4q^{3}-16q^{-3}+10q^{2}+21q^{-2}-14q-20q^{-1}+19$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{6}\left(-z^{2}\right)-a^{6}+2a^{4}z^{4}+3a^{4}z^{2}+a^{4}-a^{2}z^{6}-a^{2}z^{4}+z^{4}a^{-2}+a^{2}z^{2}+a^{2}z^{-2}+z^{2}a^{-2}+a^{-2}z^{-2}+3a^{2}+2a^{-2}-z^{6}-2z^{4}-4z^{2}-2z^{-2}-5$ (db)
Kauffman polynomial	$a^{7}z^{7}-4a^{7}z^{5}+5a^{7}z^{3}-2a^{7}z+3a^{6}z^{8}-11a^{6}z^{6}+13a^{6}z^{4}-7a^{6}z^{2}+2a^{6}+4a^{5}z^{9}-11a^{5}z^{7}+5a^{5}z^{5}+5a^{5}z^{3}-3a^{5}z+2a^{4}z^{10}+5a^{4}z^{8}-33a^{4}z^{6}+41a^{4}z^{4}+z^{4}a^{-4}-17a^{4}z^{2}+2a^{4}+12a^{3}z^{9}-31a^{3}z^{7}+21a^{3}z^{5}+4z^{5}a^{-3}-6a^{3}z^{3}+3a^{3}z+2a^{2}z^{10}+15a^{2}z^{8}-48a^{2}z^{6}+10z^{6}a^{-2}+34a^{2}z^{4}-10z^{4}a^{-2}-a^{2}z^{2}+6z^{2}a^{-2}+a^{2}z^{-2}+a^{-2}z^{-2}-6a^{2}-4a^{-2}+8az^{9}-5az^{7}+14z^{7}a^{-1}-10az^{5}-18z^{5}a^{-1}-2az^{3}+4z^{3}a^{-1}+7az+3za^{-1}-2az^{-1}-2a^{-1}z^{-1}+13z^{8}-16z^{6}-5z^{4}+15z^{2}+2z^{-2}-9$ (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 9                       1 1 7                     3   -3 5                   7 1   6 3                 7 3     -4 1               12 7       5 -1             12 11         -1 -3           9 8           1 -5         7 12             5 -7       5 9               -4 -9     2 7                 5 -11   1 5                   -4 -13   2                     2 -15 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-4$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-3$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-2$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=-1$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=0$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{12}$ $r=1$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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