L11n332

L11n331

L11n333

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n332 at Knotilus!

Link Presentations

[edit Notes on L11n332's Link Presentations]

Planar diagram presentation	X6172 X11,16,12,17 X3849 X17,2,18,3 X5,14,6,15 X18,7,19,8 X15,12,16,5 X13,20,14,21 X9,13,10,22 X21,11,22,10 X4,19,1,20
Gauss code	{1, 4, -3, -11}, {-5, -1, 6, 3, -9, 10, -2, 7}, {-8, 5, -7, 2, -4, -6, 11, 8, -10, 9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(v+w-1)(vw-v-w)(uvw-1)}{{\sqrt {u}}v^{3/2}w^{3/2}}}$ (db)
Jones polynomial	$-q^{-9}+2q^{-8}-3q^{-7}+7q^{-6}-5q^{-5}+6q^{-4}-5q^{-3}+4q^{-2}-2q^{-1}+1$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$-a^{10}+z^{4}a^{8}+4z^{2}a^{8}+a^{8}z^{-2}+2a^{8}-z^{6}a^{6}-4z^{4}a^{6}-4z^{2}a^{6}-2a^{6}z^{-2}-3a^{6}-z^{6}a^{4}-3z^{4}a^{4}+a^{4}z^{-2}+a^{4}+z^{4}a^{2}+3z^{2}a^{2}+a^{2}$ (db)
Kauffman polynomial	$z^{3}a^{11}-2za^{11}+2z^{4}a^{10}-4z^{2}a^{10}+3a^{10}+z^{7}a^{9}-4z^{5}a^{9}+10z^{3}a^{9}-7za^{9}+2z^{8}a^{8}-10z^{6}a^{8}+22z^{4}a^{8}-19z^{2}a^{8}-a^{8}z^{-2}+11a^{8}+z^{9}a^{7}-2z^{7}a^{7}-3z^{5}a^{7}+14z^{3}a^{7}-12za^{7}+2a^{7}z^{-1}+4z^{8}a^{6}-17z^{6}a^{6}+27z^{4}a^{6}-22z^{2}a^{6}-2a^{6}z^{-2}+11a^{6}+z^{9}a^{5}-z^{7}a^{5}-6z^{5}a^{5}+10z^{3}a^{5}-8za^{5}+2a^{5}z^{-1}+2z^{8}a^{4}-6z^{6}a^{4}+3z^{4}a^{4}-3z^{2}a^{4}-a^{4}z^{-2}+3a^{4}+2z^{7}a^{3}-7z^{5}a^{3}+5z^{3}a^{3}-za^{3}+z^{6}a^{2}-4z^{4}a^{2}+4z^{2}a^{2}-a^{2}$ (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 χ 1                   1 1 -1                 1   -1 -3               3 1   2 -5             3 2     -1 -7           3 2       1 -9       1 3 3         1 -11       5 3           2 -13     1 5             4 -15   1 2               -1 -17   1                 1 -19 1                   -1

Integral Khovanov Homology (db, data source)

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

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.

L11n331 L11n333