L11a427

L11a426

L11a428

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a427 at Knotilus!

Link Presentations

[edit Notes on L11a427's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X14,5,15,6 X20,7,21,8 X8,19,9,20 X16,12,17,11 X10,13,5,14 X22,18,11,17 X18,22,19,21 X2,9,3,10 X4,16,1,15
Gauss code	{1, -10, 2, -11}, {3, -1, 4, -5, 10, -7}, {6, -2, 7, -3, 11, -6, 8, -9, 5, -4, 9, -8}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {2t(1)t(3)^{2}t(2)^{2}-2t(3)^{2}t(2)^{2}+2t(1)t(2)^{2}-4t(1)t(3)t(2)^{2}+3t(3)t(2)^{2}-3t(1)t(3)^{2}t(2)+4t(3)^{2}t(2)-4t(1)t(2)+7t(1)t(3)t(2)-7t(3)t(2)+3t(2)-2t(3)^{2}+2t(1)-3t(1)t(3)+4t(3)-2}{{\sqrt {t(1)}}t(2)t(3)}}$ (db)
Jones polynomial	$-q^{4}+4q^{3}-7q^{2}+11q-15+17q^{-1}-16q^{-2}+16q^{-3}-10q^{-4}+7q^{-5}-3q^{-6}+q^{-7}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^{6}z^{2}+a^{6}z^{-2}+a^{6}-2a^{4}z^{4}-4a^{4}z^{2}-2a^{4}z^{-2}-4a^{4}+a^{2}z^{6}+2a^{2}z^{4}-z^{4}a^{-2}+3a^{2}z^{2}+a^{2}z^{-2}-z^{2}a^{-2}+4a^{2}+a^{-2}+z^{6}+2z^{4}-2$ (db)
Kauffman polynomial	$a^{8}z^{4}-a^{8}z^{2}+3a^{7}z^{5}-2a^{7}z^{3}+6a^{6}z^{6}-8a^{6}z^{4}+8a^{6}z^{2}+a^{6}z^{-2}-4a^{6}+7a^{5}z^{7}-6a^{5}z^{5}-a^{5}z^{3}+4a^{5}z-2a^{5}z^{-1}+7a^{4}z^{8}-6a^{4}z^{6}-7a^{4}z^{4}+12a^{4}z^{2}+2a^{4}z^{-2}-6a^{4}+5a^{3}z^{9}-2a^{3}z^{7}+z^{7}a^{-3}-12a^{3}z^{5}-3z^{5}a^{-3}+9a^{3}z^{3}+2z^{3}a^{-3}-2a^{3}z^{-1}+2a^{2}z^{10}+6a^{2}z^{8}+4z^{8}a^{-2}-22a^{2}z^{6}-15z^{6}a^{-2}+9a^{2}z^{4}+15z^{4}a^{-2}+5a^{2}z^{2}-2z^{2}a^{-2}+a^{2}z^{-2}-3a^{2}-a^{-2}+10az^{9}+5z^{9}a^{-1}-27az^{7}-17z^{7}a^{-1}+15az^{5}+15z^{5}a^{-1}+4az^{3}-2z^{3}a^{-1}-6az-2za^{-1}+2z^{10}+3z^{8}-25z^{6}+22z^{4}-1$ (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                     3   3 5                   4 1   -3 3                 7 3     4 1               8 4       -4 -1             9 7         2 -3           9 10           1 -5         7 7             0 -7       4 10               6 -9     3 6                 -3 -11     4                   4 -13 1 3                     -2 -15 1                       1

Integral Khovanov Homology (db, data source)

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

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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