L11n296

L11n295

L11n297

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n296 at Knotilus!

Link Presentations

[edit Notes on L11n296's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X15,20,16,21 X14,8,15,7 X10,22,5,21 X18,11,19,12 X9,17,10,16 X22,17,11,18 X19,9,20,8 X2536 X4,14,1,13
Gauss code	{1, -10, 2, -11}, {10, -1, 4, 9, -7, -5}, {6, -2, 11, -4, -3, 7, 8, -6, -9, 3, 5, -8}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {uv^{2}w+uvw^{4}-uvw^{3}+uvw^{2}-2uvw+uv-uw^{4}+uw^{3}-v^{2}w+v^{2}-vw^{4}+2vw^{3}-vw^{2}+vw-v-w^{3}}{{\sqrt {u}}vw^{2}}}$ (db)
Jones polynomial	$q^{6}-2q^{5}+4q^{4}-4q^{3}+7q^{2}-5q+6-4q^{-1}+2q^{-2}-q^{-3}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$z^{4}a^{-4}+3z^{2}a^{-4}+2a^{-4}z^{-2}+3a^{-4}-z^{6}a^{-2}-5z^{4}a^{-2}-a^{2}z^{2}-10z^{2}a^{-2}-a^{2}z^{-2}-5a^{-2}z^{-2}-2a^{2}-10a^{-2}+2z^{4}+7z^{2}+4z^{-2}+9$ (db)
Kauffman polynomial	$z^{9}a^{-1}+z^{9}a^{-3}+5z^{8}a^{-2}+3z^{8}a^{-4}+2z^{8}+az^{7}-z^{7}a^{-1}+2z^{7}a^{-5}-25z^{6}a^{-2}-14z^{6}a^{-4}+z^{6}a^{-6}-10z^{6}-4az^{5}-10z^{5}a^{-1}-13z^{5}a^{-3}-7z^{5}a^{-5}+2a^{2}z^{4}+48z^{4}a^{-2}+23z^{4}a^{-4}-4z^{4}a^{-6}+23z^{4}+a^{3}z^{3}+12az^{3}+30z^{3}a^{-1}+23z^{3}a^{-3}+4z^{3}a^{-5}-3a^{2}z^{2}-44z^{2}a^{-2}-22z^{2}a^{-4}+3z^{2}a^{-6}-22z^{2}-2a^{3}z-13az-27za^{-1}-16za^{-3}+2a^{2}+20a^{-2}+10a^{-4}+13+a^{3}z^{-1}+5az^{-1}+9a^{-1}z^{-1}+5a^{-3}z^{-1}-a^{2}z^{-2}-5a^{-2}z^{-2}-2a^{-4}z^{-2}-4z^{-2}$ (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   \ -3 -2 -1 0 1 2 3 4 5 6 χ 13                   1 1 11                 2 1 -1 9               2     2 7             2 2     0 5           5 2       3 3         1 3         2 1       5 4           1 -1     1 3             2 -3   1 3               -2 -5   1                 1 -7 1                   -1

Integral Khovanov Homology (db, data source)

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

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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