L11a248

L11a247

L11a249

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a248 at Knotilus!

Link Presentations

[edit Notes on L11a248's Link Presentations]

Planar diagram presentation	X10,1,11,2 X12,4,13,3 X22,12,9,11 X2,9,3,10 X4,22,5,21 X14,5,15,6 X20,13,21,14 X6,19,7,20 X16,8,17,7 X18,16,19,15 X8,18,1,17
Gauss code	{1, -4, 2, -5, 6, -8, 9, -11}, {4, -1, 3, -2, 7, -6, 10, -9, 11, -10, 8, -7, 5, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(u-1)(v-1)\left(u^{2}v^{2}-2u^{2}v+u^{2}-2uv^{2}+6uv-2u+v^{2}-2v+1\right)}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$-q^{13/2}+5q^{11/2}-10q^{9/2}+15q^{7/2}-21q^{5/2}+23q^{3/2}-23{\sqrt {q}}+{\frac {19}{\sqrt {q}}}-{\frac {14}{q^{3/2}}}+{\frac {8}{q^{5/2}}}-{\frac {4}{q^{7/2}}}+{\frac {1}{q^{9/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{3}a^{-5}+2z^{5}a^{-3}-a^{3}z^{3}+4z^{3}a^{-3}-a^{3}z+3za^{-3}-z^{7}a^{-1}+2az^{5}-4z^{5}a^{-1}+5az^{3}-9z^{3}a^{-1}+5az-7za^{-1}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-z^{10}a^{-2}-z^{10}-4az^{9}-9z^{9}a^{-1}-5z^{9}a^{-3}-6a^{2}z^{8}-21z^{8}a^{-2}-10z^{8}a^{-4}-17z^{8}-4a^{3}z^{7}-5az^{7}-5z^{7}a^{-1}-14z^{7}a^{-3}-10z^{7}a^{-5}-a^{4}z^{6}+12a^{2}z^{6}+41z^{6}a^{-2}+9z^{6}a^{-4}-5z^{6}a^{-6}+40z^{6}+10a^{3}z^{5}+30az^{5}+47z^{5}a^{-1}+43z^{5}a^{-3}+15z^{5}a^{-5}-z^{5}a^{-7}+2a^{4}z^{4}-6a^{2}z^{4}-19z^{4}a^{-2}+4z^{4}a^{-4}+5z^{4}a^{-6}-26z^{4}-8a^{3}z^{3}-30az^{3}-46z^{3}a^{-1}-30z^{3}a^{-3}-6z^{3}a^{-5}-a^{4}z^{2}-3z^{2}a^{-4}+4z^{2}+2a^{3}z+10az+14za^{-1}+6za^{-3}+1-az^{-1}-a^{-1}z^{-1}$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 14                       1 1 12                     4   -4 10                   6 1   5 8                 9 4     -5 6               12 6       6 4             11 9         -2 2           12 12           0 0         9 13             4 -2       5 10               -5 -4     3 9                 6 -6   1 5                   -4 -8   3                     3 -10 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=0$ $i=2$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-2$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=0$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{12}$ $r=1$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=2$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=3$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=4$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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