L11n248

L11n247

L11n249

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n248 at Knotilus!

Link Presentations

[edit Notes on L11n248's Link Presentations]

Planar diagram presentation	X12,1,13,2 X8493 X14,6,15,5 X18,8,19,7 X20,9,21,10 X10,11,1,12 X6,14,7,13 X4,18,5,17 X15,11,16,22 X2,19,3,20 X21,17,22,16
Gauss code	{1, -10, 2, -8, 3, -7, 4, -2, 5, -6}, {6, -1, 7, -3, -9, 11, 8, -4, 10, -5, -11, 9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(u-1)(v-1)\left(u^{2}v^{2}-u^{2}v+u^{2}-uv^{2}+3uv-u+v^{2}-v+1\right)}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$-6q^{9/2}+10q^{7/2}-{\frac {1}{q^{7/2}}}-14q^{5/2}+{\frac {4}{q^{5/2}}}+15q^{3/2}-{\frac {9}{q^{3/2}}}+2q^{11/2}-15{\sqrt {q}}+{\frac {12}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$za^{-5}+z^{5}a^{-3}+z^{3}a^{-3}-z^{7}a^{-1}+az^{5}-4z^{5}a^{-1}+2az^{3}-6z^{3}a^{-1}+2az-3za^{-1}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$3z^{4}a^{-6}-3z^{2}a^{-6}+z^{7}a^{-5}+5z^{5}a^{-5}-7z^{3}a^{-5}+2za^{-5}+3z^{8}a^{-4}-z^{6}a^{-4}+z^{4}a^{-4}-z^{2}a^{-4}+2z^{9}a^{-3}+4z^{7}a^{-3}+a^{3}z^{5}-7z^{5}a^{-3}-a^{3}z^{3}+2z^{3}a^{-3}+10z^{8}a^{-2}+4a^{2}z^{6}-14z^{6}a^{-2}-5a^{2}z^{4}+3z^{4}a^{-2}+a^{2}z^{2}+2z^{2}a^{-2}+2z^{9}a^{-1}+8az^{7}+11z^{7}a^{-1}-15az^{5}-28z^{5}a^{-1}+9az^{3}+19z^{3}a^{-1}-4az-6za^{-1}+az^{-1}+a^{-1}z^{-1}+7z^{8}-9z^{6}+z^{2}-1$ (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   \ -4 -3 -2 -1 0 1 2 3 4 5 χ 12                   2 -2 10                 4   4 8               6 2   -4 6             8 4     4 4           7 6       -1 2         8 8         0 0       6 9           3 -2     3 6             -3 -4   1 6               5 -6   3                 -3 -8 1                   1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=0$ $i=2$ $r=-4$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-1$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=0$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{8}$ $r=1$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=2$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=3$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}} ${\mathbb {Z} }^{6}$ $r=4$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=5$ ${\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$

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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