L11a251

L11a250

L11a252

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a251 at Knotilus!

Link Presentations

[edit Notes on L11a251's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(u-1)(v-1)^{3}(uv+1)^{2}}{u^{3/2}v^{5/2}}}$ (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	$-z^{7}a^{-3}-4z^{5}a^{-3}-4z^{3}a^{-3}+z^{9}a^{-1}-az^{7}+6z^{7}a^{-1}-4az^{5}+12z^{5}a^{-1}-4az^{3}+8z^{3}a^{-1}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-3z^{10}a^{-2}-3z^{10}-7az^{9}-14z^{9}a^{-1}-7z^{9}a^{-3}-7a^{2}z^{8}-4z^{8}a^{-2}-8z^{8}a^{-4}-3z^{8}-4a^{3}z^{7}+18az^{7}+41z^{7}a^{-1}+12z^{7}a^{-3}-7z^{7}a^{-5}-a^{4}z^{6}+19a^{2}z^{6}+20z^{6}a^{-2}+12z^{6}a^{-4}-4z^{6}a^{-6}+24z^{6}+10a^{3}z^{5}-15az^{5}-48z^{5}a^{-1}-11z^{5}a^{-3}+11z^{5}a^{-5}-z^{5}a^{-7}+2a^{4}z^{4}-13a^{2}z^{4}-23z^{4}a^{-2}-4z^{4}a^{-4}+6z^{4}a^{-6}-28z^{4}-4a^{3}z^{3}+6az^{3}+19z^{3}a^{-1}+5z^{3}a^{-3}-3z^{3}a^{-5}+z^{3}a^{-7}+3a^{2}z^{2}+6z^{2}a^{-2}-z^{2}a^{-6}+8z^{2}+1-az^{-1}-a^{-1}z^{-1}$ (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                     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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