L11a376

L11a375

L11a377

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a376 at Knotilus!

Link Presentations

[edit Notes on L11a376's Link Presentations]

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(2)t(1)-t(1)+1)(t(1)t(2)-t(2)+1)\left(t(2)t(1)^{2}+t(2)^{2}t(1)-t(2)t(1)+t(1)+t(2)\right)}{t(1)^{2}t(2)^{2}}}$ (db)
Jones polynomial	${\frac {14}{q^{9/2}}}-{\frac {15}{q^{7/2}}}+{\frac {12}{q^{5/2}}}+q^{3/2}-{\frac {10}{q^{3/2}}}-{\frac {1}{q^{19/2}}}+{\frac {2}{q^{17/2}}}-{\frac {5}{q^{15/2}}}+{\frac {9}{q^{13/2}}}-{\frac {12}{q^{11/2}}}-3{\sqrt {q}}+{\frac {6}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$z^{5}a^{7}+4z^{3}a^{7}+5za^{7}+a^{7}z^{-1}-z^{7}a^{5}-5z^{5}a^{5}-10z^{3}a^{5}-8za^{5}-a^{5}z^{-1}-z^{7}a^{3}-4z^{5}a^{3}-5z^{3}a^{3}-2za^{3}+z^{5}a+3z^{3}a+2za$ (db)
Kauffman polynomial	$-z^{5}a^{11}+3z^{3}a^{11}-2za^{11}-2z^{6}a^{10}+4z^{4}a^{10}-z^{2}a^{10}-3z^{7}a^{9}+5z^{5}a^{9}-2z^{3}a^{9}+za^{9}-4z^{8}a^{8}+10z^{6}a^{8}-15z^{4}a^{8}+9z^{2}a^{8}-3z^{9}a^{7}+5z^{7}a^{7}-6z^{5}a^{7}+3z^{3}a^{7}-4za^{7}+a^{7}z^{-1}-z^{10}a^{6}-5z^{8}a^{6}+19z^{6}a^{6}-29z^{4}a^{6}+14z^{2}a^{6}-a^{6}-6z^{9}a^{5}+15z^{7}a^{5}-18z^{5}a^{5}+13z^{3}a^{5}-8za^{5}+a^{5}z^{-1}-z^{10}a^{4}-5z^{8}a^{4}+18z^{6}a^{4}-18z^{4}a^{4}+7z^{2}a^{4}-3z^{9}a^{3}+4z^{7}a^{3}+3z^{5}a^{3}-2z^{3}a^{3}+za^{3}-4z^{8}a^{2}+10z^{6}a^{2}-5z^{4}a^{2}+z^{2}a^{2}-3z^{7}a+9z^{5}a-7z^{3}a+2za-z^{6}+3z^{4}-2z^{2}$ (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             8 5         3 -8           6 7           1 -10         6 8             -2 -12       3 6               3 -14     2 6                 -4 -16   1 4                   3 -18   1                     -1 -20 1                       1

Integral Khovanov Homology (db, data source)

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

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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