L11a514

L11a513

L11a515

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a514 at Knotilus!

Link Presentations

[edit Notes on L11a514's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {u^{2}v^{2}w^{2}-u^{2}v^{2}w+u^{2}vw^{3}-3u^{2}vw^{2}+2u^{2}vw-u^{2}w^{3}+2u^{2}w^{2}-2uv^{2}w^{2}+3uv^{2}w-uv^{2}-2uvw^{3}+5uvw^{2}-5uvw+2uv+uw^{3}-3uw^{2}+2uw-2v^{2}w+v^{2}-2vw^{2}+3vw-v+w^{2}-w}{uvw^{3/2}}}$ (db)
Jones polynomial	$-q^{4}+3q^{3}-5q^{2}+10q-12+15q^{-1}-15q^{-2}+14q^{-3}-10q^{-4}+7q^{-5}-3q^{-6}+q^{-7}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^{6}z^{2}+a^{6}-2a^{4}z^{4}-3a^{4}z^{2}+a^{4}z^{-2}+a^{4}+a^{2}z^{6}+a^{2}z^{4}-z^{4}a^{-2}-4a^{2}z^{2}-2a^{2}z^{-2}-2z^{2}a^{-2}-6a^{2}+z^{6}+3z^{4}+4z^{2}+z^{-2}+4$ (db)
Kauffman polynomial	$a^{2}z^{10}+z^{10}+3a^{3}z^{9}+6az^{9}+3z^{9}a^{-1}+6a^{4}z^{8}+8a^{2}z^{8}+3z^{8}a^{-2}+5z^{8}+7a^{5}z^{7}+6a^{3}z^{7}-11az^{7}-9z^{7}a^{-1}+z^{7}a^{-3}+6a^{6}z^{6}-5a^{4}z^{6}-24a^{2}z^{6}-13z^{6}a^{-2}-26z^{6}+3a^{7}z^{5}-8a^{5}z^{5}-25a^{3}z^{5}-5az^{5}+5z^{5}a^{-1}-4z^{5}a^{-3}+a^{8}z^{4}-8a^{6}z^{4}-a^{4}z^{4}+23a^{2}z^{4}+18z^{4}a^{-2}+33z^{4}-2a^{7}z^{3}+4a^{5}z^{3}+22a^{3}z^{3}+14az^{3}+2z^{3}a^{-1}+4z^{3}a^{-3}-a^{8}z^{2}+7a^{6}z^{2}-a^{4}z^{2}-21a^{2}z^{2}-8z^{2}a^{-2}-20z^{2}-9a^{3}z-9az-2a^{6}+3a^{4}+11a^{2}+7+2a^{3}z^{-1}+2az^{-1}-a^{4}z^{-2}-2a^{2}z^{-2}-z^{-2}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 9                       1 -1 7                     2   2 5                   3 1   -2 3                 7 2     5 1               6 4       -2 -1             9 6         3 -3           8 8           0 -5         6 7             -1 -7       4 8               4 -9     3 6                 -3 -11     4                   4 -13 1 3                     -2 -15 1                       1

Integral Khovanov Homology (db, data source)

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

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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