L11a416

L11a415

L11a417

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a416 at Knotilus!

Link Presentations

[edit Notes on L11a416's Link Presentations]

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {t(1)t(3)^{2}t(2)^{2}-2t(3)^{2}t(2)^{2}+2t(1)t(2)^{2}-3t(1)t(3)t(2)^{2}+4t(3)t(2)^{2}-2t(2)^{2}-3t(1)t(3)^{2}t(2)+4t(3)^{2}t(2)-4t(1)t(2)+8t(1)t(3)t(2)-8t(3)t(2)+3t(2)+2t(1)t(3)^{2}-2t(3)^{2}+2t(1)-4t(1)t(3)+3t(3)-1}{{\sqrt {t(1)}}t(2)t(3)}}$ (db)
Jones polynomial	$-q^{-8}+3q^{-7}-6q^{-6}+12q^{-5}-15q^{-4}+q^{3}+19q^{-3}-4q^{2}-18q^{-2}+8q+17q^{-1}-12$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-a^{8}+3a^{6}z^{2}+a^{6}z^{-2}+3a^{6}-3a^{4}z^{4}-5a^{4}z^{2}-2a^{4}z^{-2}-5a^{4}+a^{2}z^{6}+2a^{2}z^{4}+4a^{2}z^{2}+a^{2}z^{-2}+z^{2}a^{-2}+3a^{2}-2z^{4}-2z^{2}$ (db)
Kauffman polynomial	$a^{9}z^{5}-2a^{9}z^{3}+a^{9}z+3a^{8}z^{6}-6a^{8}z^{4}+5a^{8}z^{2}-2a^{8}+4a^{7}z^{7}-3a^{7}z^{5}-3a^{7}z^{3}+3a^{7}z+4a^{6}z^{8}+2a^{6}z^{6}-15a^{6}z^{4}+18a^{6}z^{2}+a^{6}z^{-2}-8a^{6}+3a^{5}z^{9}+4a^{5}z^{7}-9a^{5}z^{5}+a^{5}z^{3}+5a^{5}z-2a^{5}z^{-1}+a^{4}z^{10}+9a^{4}z^{8}-13a^{4}z^{6}-4a^{4}z^{4}+16a^{4}z^{2}+2a^{4}z^{-2}-9a^{4}+7a^{3}z^{9}-5a^{3}z^{7}-8a^{3}z^{5}+4a^{3}z^{3}+3a^{3}z-2a^{3}z^{-1}+a^{2}z^{10}+11a^{2}z^{8}-27a^{2}z^{6}+z^{6}a^{-2}+15a^{2}z^{4}-2z^{4}a^{-2}+a^{2}z^{2}+z^{2}a^{-2}+a^{2}z^{-2}-4a^{2}+4az^{9}-az^{7}+4z^{7}a^{-1}-13az^{5}-10z^{5}a^{-1}+8az^{3}+6z^{3}a^{-1}+6z^{8}-14z^{6}+8z^{4}-z^{2}$ (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   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 7                       1 1 5                     3   -3 3                   5 1   4 1                 7 3     -4 -1               10 5       5 -3             9 8         -1 -5           10 9           1 -7         7 11             4 -9       5 8               -3 -11     2 8                 6 -13   1 4                   -3 -15   2                     2 -17 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-3$ $i=-1$ $r=-7$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-4$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{5}$ $r=-3$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-2$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{10}$ $r=-1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=0$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{10}$ $r=1$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=4$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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