L11a247

L11a246

L11a248

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a247 at Knotilus!

Link Presentations

[edit Notes on L11a247's Link Presentations]

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

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(u-1)^{3}(v-1)^{3}}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$-q^{13/2}+4q^{11/2}-8q^{9/2}+13q^{7/2}-18q^{5/2}+20q^{3/2}-21{\sqrt {q}}+{\frac {17}{\sqrt {q}}}-{\frac {13}{q^{3/2}}}+{\frac {8}{q^{5/2}}}-{\frac {4}{q^{7/2}}}+{\frac {1}{q^{9/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-1}+2az^{5}-4z^{5}a^{-1}+2z^{5}a^{-3}-a^{3}z^{3}+5az^{3}-8z^{3}a^{-1}+5z^{3}a^{-3}-z^{3}a^{-5}-a^{3}z+4az-6za^{-1}+4za^{-3}-za^{-5}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{5}a^{-7}-z^{3}a^{-7}+4z^{6}a^{-6}-6z^{4}a^{-6}+2z^{2}a^{-6}+7z^{7}a^{-5}-11z^{5}a^{-5}+6z^{3}a^{-5}-2za^{-5}+7z^{8}a^{-4}+a^{4}z^{6}-6z^{6}a^{-4}-2a^{4}z^{4}-3z^{4}a^{-4}+a^{4}z^{2}+3z^{2}a^{-4}+4z^{9}a^{-3}+4a^{3}z^{7}+8z^{7}a^{-3}-10a^{3}z^{5}-28z^{5}a^{-3}+7a^{3}z^{3}+25z^{3}a^{-3}-2a^{3}z-8za^{-3}+z^{10}a^{-2}+6a^{2}z^{8}+15z^{8}a^{-2}-13a^{2}z^{6}-32z^{6}a^{-2}+6a^{2}z^{4}+20z^{4}a^{-2}-2z^{2}a^{-2}+4az^{9}+8z^{9}a^{-1}+2az^{7}-z^{7}a^{-1}-24az^{5}-30z^{5}a^{-1}+24az^{3}+35z^{3}a^{-1}-8az-12za^{-1}+az^{-1}+a^{-1}z^{-1}+z^{10}+14z^{8}-36z^{6}+25z^{4}-4z^{2}-1$ (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   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 14                       1 1 12                     3   -3 10                   5 1   4 8                 8 3     -5 6               10 5       5 4             10 8         -2 2           11 10           1 0         8 12             4 -2       5 9               -4 -4     3 8                 5 -6   1 5                   -4 -8   3                     3 -10 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=0$ $i=2$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-2$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=0$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{11}$ $r=1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=2$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=3$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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