L11a456

L11a455

L11a457

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a456 at Knotilus!

Link Presentations

[edit Notes on L11a456's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {uv^{3}w^{2}-uv^{3}w+uv^{2}w^{3}-4uv^{2}w^{2}+5uv^{2}w-uv^{2}-2uvw^{3}+5uvw^{2}-4uvw+uv+uw^{3}-2uw^{2}+uw-v^{3}w^{2}+2v^{3}w-v^{3}-v^{2}w^{3}+4v^{2}w^{2}-5v^{2}w+2v^{2}+vw^{3}-5vw^{2}+4vw-v+w^{2}-w}{{\sqrt {u}}v^{3/2}w^{3/2}}}$ (db)
Jones polynomial	$q^{6}-3q^{5}-q^{-5}+7q^{4}+3q^{-4}-11q^{3}-7q^{-3}+17q^{2}+13q^{-2}-18q-16q^{-1}+19$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$z^{4}a^{-4}-a^{4}z^{2}+2z^{2}a^{-4}+a^{-4}z^{-2}-a^{4}+2a^{-4}-z^{6}a^{-2}+2a^{2}z^{4}-3z^{4}a^{-2}+3a^{2}z^{2}-6z^{2}a^{-2}-2a^{-2}z^{-2}+2a^{2}-5a^{-2}-z^{6}-z^{4}+z^{2}+z^{-2}+2$ (db)
Kauffman polynomial	$z^{6}a^{-6}-3z^{4}a^{-6}+2z^{2}a^{-6}+3z^{7}a^{-5}+a^{5}z^{5}-8z^{5}a^{-5}-2a^{5}z^{3}+5z^{3}a^{-5}+a^{5}z+5z^{8}a^{-4}+3a^{4}z^{6}-14z^{6}a^{-4}-5a^{4}z^{4}+16z^{4}a^{-4}+3a^{4}z^{2}-13z^{2}a^{-4}-a^{-4}z^{-2}-a^{4}+6a^{-4}+4z^{9}a^{-3}+5a^{3}z^{7}-5z^{7}a^{-3}-6a^{3}z^{5}-3z^{5}a^{-3}+a^{3}z^{3}+6z^{3}a^{-3}+a^{3}z-5za^{-3}+2a^{-3}z^{-1}+z^{10}a^{-2}+6a^{2}z^{8}+12z^{8}a^{-2}-7a^{2}z^{6}-41z^{6}a^{-2}+3a^{2}z^{4}+55z^{4}a^{-2}-a^{2}z^{2}-39z^{2}a^{-2}-2a^{-2}z^{-2}+13a^{-2}+4az^{9}+8z^{9}a^{-1}+2az^{7}-11z^{7}a^{-1}-12az^{5}+10az^{3}+8z^{3}a^{-1}-3az-8za^{-1}+2a^{-1}z^{-1}+z^{10}+13z^{8}-36z^{6}+44z^{4}-28z^{2}-z^{-2}+9$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 13                       1 1 11                     2   -2 9                   5 1   4 7                 7 3     -4 5               10 4       6 3             8 7         -1 1           11 10           1 -1         8 11             3 -3       5 8               -3 -5     2 8                 6 -7   1 5                   -4 -9   2                     2 -11 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-2$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-1$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=0$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{11}$ $r=1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=2$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=3$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{5}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

L11a455 L11a457