L11a258

L11a257

L11a259

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a258 at Knotilus!

Link Presentations

[edit Notes on L11a258's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {u^{3}v^{4}-2u^{3}v^{3}+2u^{3}v^{2}+u^{2}v^{5}-3u^{2}v^{4}+6u^{2}v^{3}-5u^{2}v^{2}+2u^{2}v+2uv^{4}-5uv^{3}+6uv^{2}-3uv+u+2v^{3}-2v^{2}+v}{u^{3/2}v^{5/2}}}$ (db)
Jones polynomial	$q^{3/2}-3{\sqrt {q}}+{\frac {6}{\sqrt {q}}}-{\frac {10}{q^{3/2}}}+{\frac {12}{q^{5/2}}}-{\frac {14}{q^{7/2}}}+{\frac {14}{q^{9/2}}}-{\frac {12}{q^{11/2}}}+{\frac {8}{q^{13/2}}}-{\frac {5}{q^{15/2}}}+{\frac {2}{q^{17/2}}}-{\frac {1}{q^{19/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$z^{5}a^{7}+4z^{3}a^{7}+5za^{7}+2a^{7}z^{-1}-z^{7}a^{5}-5z^{5}a^{5}-10z^{3}a^{5}-9za^{5}-3a^{5}z^{-1}-z^{7}a^{3}-4z^{5}a^{3}-5z^{3}a^{3}-2za^{3}+a^{3}z^{-1}+z^{5}a+3z^{3}a+2za$ (db)
Kauffman polynomial	$-z^{5}a^{11}+3z^{3}a^{11}-2za^{11}-2z^{6}a^{10}+4z^{4}a^{10}-z^{2}a^{10}-3z^{7}a^{9}+6z^{5}a^{9}-5z^{3}a^{9}+3za^{9}-3z^{8}a^{8}+4z^{6}a^{8}-2z^{4}a^{8}-3z^{9}a^{7}+7z^{7}a^{7}-14z^{5}a^{7}+13z^{3}a^{7}-8za^{7}+2a^{7}z^{-1}-z^{10}a^{6}-4z^{8}a^{6}+15z^{6}a^{6}-23z^{4}a^{6}+12z^{2}a^{6}-3a^{6}-6z^{9}a^{5}+17z^{7}a^{5}-26z^{5}a^{5}+24z^{3}a^{5}-14za^{5}+3a^{5}z^{-1}-z^{10}a^{4}-5z^{8}a^{4}+20z^{6}a^{4}-25z^{4}a^{4}+15z^{2}a^{4}-3a^{4}-3z^{9}a^{3}+4z^{7}a^{3}+4z^{5}a^{3}-4z^{3}a^{3}+za^{3}+a^{3}z^{-1}-4z^{8}a^{2}+10z^{6}a^{2}-5z^{4}a^{2}+2z^{2}a^{2}-a^{2}-3z^{7}a+9z^{5}a-7z^{3}a+2za-z^{6}+3z^{4}-2z^{2}$ (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   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 4                       1 -1 2                     2   2 0                   4 1   -3 -2                 6 2     4 -4               7 5       -2 -6             7 5         2 -8           7 7           0 -10         5 7             -2 -12       3 7               4 -14     2 5                 -3 -16     3                   3 -18 1 2                     -1 -20 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-4$ $i=-2$ $r=-8$ ${\mathbb {Z} }$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }^{2}$ $r=-6$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-5$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-4$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-3$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-2$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-1$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=0$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{6}$ $r=1$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=2$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=3$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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