L11a15

L11a14

L11a16

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a15 at Knotilus!

Link Presentations

[edit Notes on L11a15's Link Presentations]

Planar diagram presentation	X6172 X12,7,13,8 X4,13,1,14 X18,10,19,9 X8493 X14,6,15,5 X22,16,5,15 X20,18,21,17 X16,22,17,21 X10,20,11,19 X2,12,3,11
Gauss code	{1, -11, 5, -3}, {6, -1, 2, -5, 4, -10, 11, -2, 3, -6, 7, -9, 8, -4, 10, -8, 9, -7}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(u-1)(v-1)\left(3v^{2}-5v+3\right)}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$-12q^{9/2}+13q^{7/2}-14q^{5/2}+{\frac {1}{q^{5/2}}}+11q^{3/2}-{\frac {3}{q^{3/2}}}-q^{17/2}+4q^{15/2}-6q^{13/2}+9q^{11/2}-9{\sqrt {q}}+{\frac {5}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{3}a^{-7}+a^{-7}z^{-1}+z^{5}a^{-5}+z^{3}a^{-5}-za^{-5}-2a^{-5}z^{-1}+z^{5}a^{-3}-za^{-3}+z^{5}a^{-1}-az^{3}+2z^{3}a^{-1}-az+3za^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{7}a^{-9}-3z^{5}a^{-9}+z^{3}a^{-9}+4z^{8}a^{-8}-16z^{6}a^{-8}+15z^{4}a^{-8}-z^{2}a^{-8}-2a^{-8}+5z^{9}a^{-7}-20z^{7}a^{-7}+21z^{5}a^{-7}-6z^{3}a^{-7}+a^{-7}z^{-1}+2z^{10}a^{-6}-21z^{6}a^{-6}+24z^{4}a^{-6}+z^{2}a^{-6}-5a^{-6}+9z^{9}a^{-5}-33z^{7}a^{-5}+36z^{5}a^{-5}-13z^{3}a^{-5}-za^{-5}+2a^{-5}z^{-1}+2z^{10}a^{-4}-11z^{6}a^{-4}+9z^{4}a^{-4}+2z^{2}a^{-4}-3a^{-4}+4z^{9}a^{-3}-8z^{7}a^{-3}+9z^{5}a^{-3}-9z^{3}a^{-3}+2za^{-3}+4z^{8}a^{-2}-2z^{6}a^{-2}+a^{2}z^{4}-4z^{4}a^{-2}-a^{2}z^{2}+z^{2}a^{-2}+a^{-2}+4z^{7}a^{-1}+3az^{5}-4az^{3}-7z^{3}a^{-1}+2az+5za^{-1}-a^{-1}z^{-1}+4z^{6}-3z^{4}$ (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   \ -3 -2 -1 0 1 2 3 4 5 6 7 8 χ 18                       1 1 16                     3   -3 14                   3 1   2 12                 6 3     -3 10               6 3       3 8             7 6         -1 6           7 6           1 4         4 7             3 2       5 7               -2 0     2 6                 4 -2   1 3                   -2 -4   2                     2 -6 1                       -1

Integral Khovanov Homology (db, data source)

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

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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