L11a515

L11a514

L11a516

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a515 at Knotilus!

Link Presentations

[edit Notes on L11a515's Link Presentations]

Planar diagram presentation	X8192 X14,4,15,3 X20,11,21,12 X18,10,19,9 X22,19,13,20 X10,14,11,13 X12,21,7,22 X16,6,17,5 X2738 X4,16,5,15 X6,18,1,17
Gauss code	{1, -9, 2, -10, 8, -11}, {9, -1, 4, -6, 3, -7}, {6, -2, 10, -8, 11, -4, 5, -3, 7, -5}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {u^{2}v^{2}w^{3}-u^{2}v^{2}w^{2}-2u^{2}vw^{3}+4u^{2}vw^{2}-u^{2}vw+u^{2}w^{3}-2u^{2}w^{2}+u^{2}w-uv^{2}w^{3}+2uv^{2}w^{2}-uv^{2}w+uvw^{3}-4uvw^{2}+4uvw-uv+uw^{2}-2uw+u-v^{2}w^{2}+2v^{2}w-v^{2}+vw^{2}-4vw+2v+w-1}{uvw^{3/2}}}$ (db)
Jones polynomial	$q^{7}-3q^{6}+6q^{5}-9q^{4}-q^{-4}+13q^{3}+3q^{-3}-13q^{2}-5q^{-2}+14q+9q^{-1}-11$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{6}a^{-4}+4z^{4}a^{-4}+5z^{2}a^{-4}+a^{-4}z^{-2}+3a^{-4}-z^{8}a^{-2}-6z^{6}a^{-2}-a^{2}z^{4}-14z^{4}a^{-2}-3a^{2}z^{2}-16z^{2}a^{-2}-2a^{-2}z^{-2}-a^{2}-8a^{-2}+2z^{6}+9z^{4}+12z^{2}+z^{-2}+6$ (db)
Kauffman polynomial	$z^{4}a^{-8}-z^{2}a^{-8}+3z^{5}a^{-7}-3z^{3}a^{-7}+5z^{6}a^{-6}-6z^{4}a^{-6}+3z^{2}a^{-6}-a^{-6}+6z^{7}a^{-5}-8z^{5}a^{-5}+4z^{3}a^{-5}+6z^{8}a^{-4}-11z^{6}a^{-4}+10z^{4}a^{-4}-6z^{2}a^{-4}-a^{-4}z^{-2}+4a^{-4}+4z^{9}a^{-3}+a^{3}z^{7}-5z^{7}a^{-3}-4a^{3}z^{5}-3z^{5}a^{-3}+4a^{3}z^{3}+8z^{3}a^{-3}-a^{3}z-6za^{-3}+2a^{-3}z^{-1}+z^{10}a^{-2}+3a^{2}z^{8}+9z^{8}a^{-2}-13a^{2}z^{6}-40z^{6}a^{-2}+17a^{2}z^{4}+56z^{4}a^{-2}-9a^{2}z^{2}-37z^{2}a^{-2}-2a^{-2}z^{-2}+2a^{2}+12a^{-2}+3az^{9}+7z^{9}a^{-1}-9az^{7}-21z^{7}a^{-1}+2az^{5}+14z^{5}a^{-1}+7az^{3}+4z^{3}a^{-1}-3az-8za^{-1}+2a^{-1}z^{-1}+z^{10}+6z^{8}-37z^{6}+56z^{4}-36z^{2}-z^{-2}+10$ (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 15                       1 1 13                     2   -2 11                   4 1   3 9                 6 3     -3 7               7 3       4 5             7 7         0 3           7 6           1 1         5 8             3 -1       4 6               -2 -3     2 6                 4 -5   1 3                   -2 -7   2                     2 -9 1                       -1

Integral Khovanov Homology (db, data source)

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

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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