L11n275

L11n274

L11n276

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n275 at Knotilus!

Link Presentations

[edit Notes on L11n275's Link Presentations]

Planar diagram presentation	X6172 X5,12,6,13 X3849 X2,14,3,13 X14,7,15,8 X9,18,10,19 X17,11,18,22 X11,21,12,20 X21,17,22,16 X15,1,16,4 X19,10,20,5
Gauss code	{1, -4, -3, 10}, {-2, -1, 5, 3, -6, 11}, {-8, 2, 4, -5, -10, 9, -7, 6, -11, 8, -9, 7}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {uvw^{4}-3uvw^{3}+3uvw^{2}-uvw+2uw^{3}-2uw^{2}+uw-v^{2}w^{3}+2v^{2}w^{2}-2v^{2}w+vw^{3}-3vw^{2}+3vw-v}{{\sqrt {u}}vw^{2}}}$ (db)
Jones polynomial	$-2q^{3}+4q^{2}-6q+9-8q^{-1}+9q^{-2}-6q^{-3}+5q^{-4}-2q^{-5}+q^{-6}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-a^{2}z^{6}+a^{4}z^{4}-5a^{2}z^{4}+3z^{4}+3a^{4}z^{2}-12a^{2}z^{2}-2z^{2}a^{-2}+10z^{2}+4a^{4}-13a^{2}-3a^{-2}+12+2a^{4}z^{-2}-5a^{2}z^{-2}-a^{-2}z^{-2}+4z^{-2}$ (db)
Kauffman polynomial	$a^{3}z^{9}+az^{9}+3a^{4}z^{8}+6a^{2}z^{8}+3z^{8}+2a^{5}z^{7}+3a^{3}z^{7}+4az^{7}+3z^{7}a^{-1}+a^{6}z^{6}-11a^{4}z^{6}-22a^{2}z^{6}+z^{6}a^{-2}-9z^{6}-6a^{5}z^{5}-20a^{3}z^{5}-23az^{5}-9z^{5}a^{-1}-4a^{6}z^{4}+15a^{4}z^{4}+36a^{2}z^{4}+2z^{4}a^{-2}+19z^{4}+3a^{5}z^{3}+28a^{3}z^{3}+43az^{3}+21z^{3}a^{-1}+3z^{3}a^{-3}+4a^{6}z^{2}-15a^{4}z^{2}-36a^{2}z^{2}-4z^{2}a^{-2}-21z^{2}-18a^{3}z-33az-19za^{-1}-4za^{-3}-a^{6}+8a^{4}+20a^{2}+3a^{-2}+15+5a^{3}z^{-1}+9az^{-1}+5a^{-1}z^{-1}+a^{-3}z^{-1}-2a^{4}z^{-2}-5a^{2}z^{-2}-a^{-2}z^{-2}-4z^{-2}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 χ 7                   2 -2 5                 2   2 3               4 2   -2 1             5 2     3 -1           5 6       1 -3         4 3         1 -5       3 6           3 -7     2 3             -1 -9     3               3 -11 1 2                 -1 -13 1                   1

Integral Khovanov Homology (db, data source)

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

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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