L11n295

L11n294

L11n296

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n295 at Knotilus!

Link Presentations

[edit Notes on L11n295's Link Presentations]

Planar diagram presentation	X6172 X12,4,13,3 X20,16,21,15 X14,8,15,7 X21,10,22,5 X11,19,12,18 X9,17,10,16 X17,11,18,22 X8,19,9,20 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 $u$, $v$, $w$, ...)	${\frac {uv^{2}w^{2}-uv^{2}w-3uvw^{2}+4uvw-2uv+uw^{2}-2uw+u-v^{2}w^{2}+2v^{2}w-v^{2}+2vw^{2}-4vw+3v+w-1}{{\sqrt {u}}vw}}$ (db)
Jones polynomial	$-q^{8}+3q^{7}-6q^{6}+8q^{5}-10q^{4}+11q^{3}-8q^{2}+8q-3+2q^{-1}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-z^{4}a^{-6}-2z^{2}a^{-6}-a^{-6}z^{-2}-2a^{-6}+z^{6}a^{-4}+4z^{4}a^{-4}+8z^{2}a^{-4}+4a^{-4}z^{-2}+8a^{-4}-3z^{4}a^{-2}-9z^{2}a^{-2}-5a^{-2}z^{-2}-10a^{-2}+2z^{2}+2z^{-2}+4$ (db)
Kauffman polynomial	$z^{5}a^{-9}-2z^{3}a^{-9}+3z^{6}a^{-8}-6z^{4}a^{-8}+z^{2}a^{-8}+5z^{7}a^{-7}-13z^{5}a^{-7}+11z^{3}a^{-7}-5za^{-7}+a^{-7}z^{-1}+4z^{8}a^{-6}-9z^{6}a^{-6}+9z^{4}a^{-6}-6z^{2}a^{-6}-a^{-6}z^{-2}+3a^{-6}+z^{9}a^{-5}+6z^{7}a^{-5}-23z^{5}a^{-5}+32z^{3}a^{-5}-18za^{-5}+5a^{-5}z^{-1}+6z^{8}a^{-4}-18z^{6}a^{-4}+30z^{4}a^{-4}-24z^{2}a^{-4}-4a^{-4}z^{-2}+12a^{-4}+z^{9}a^{-3}+2z^{7}a^{-3}-10z^{5}a^{-3}+24z^{3}a^{-3}-24za^{-3}+9a^{-3}z^{-1}+2z^{8}a^{-2}-6z^{6}a^{-2}+18z^{4}a^{-2}-25z^{2}a^{-2}-5a^{-2}z^{-2}+15a^{-2}+z^{7}a^{-1}-z^{5}a^{-1}+5z^{3}a^{-1}-11za^{-1}+5a^{-1}z^{-1}+3z^{4}-8z^{2}-2z^{-2}+7$ (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   \ -2 -1 0 1 2 3 4 5 6 7 χ 17                   1 -1 15                 2   2 13               4 1   -3 11             4 2     2 9           6 4       -2 7         5 4         1 5       4 7           3 3     4 4             0 1   1 6               5 -1 1 2                 -1 -3 2                   2

Integral Khovanov Homology (db, data source)

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

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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