L11a350

L11a349

L11a351

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a350 at Knotilus!

Link Presentations

[edit Notes on L11a350's Link Presentations]

Planar diagram presentation	X12,1,13,2 X8493 X18,14,19,13 X22,20,11,19 X20,7,21,8 X6,21,7,22 X4,15,5,16 X14,5,15,6 X16,10,17,9 X2,11,3,12 X10,18,1,17
Gauss code	{1, -10, 2, -7, 8, -6, 5, -2, 9, -11}, {10, -1, 3, -8, 7, -9, 11, -3, 4, -5, 6, -4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(2)^{3}t(1)^{3}-3t(2)^{2}t(1)^{3}+4t(2)t(1)^{3}-2t(1)^{3}-3t(2)^{3}t(1)^{2}+8t(2)^{2}t(1)^{2}-8t(2)t(1)^{2}+4t(1)^{2}+4t(2)^{3}t(1)-8t(2)^{2}t(1)+8t(2)t(1)-3t(1)-2t(2)^{3}+4t(2)^{2}-3t(2)+1}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$q^{11/2}-4q^{9/2}+9q^{7/2}-15q^{5/2}+19q^{3/2}-21{\sqrt {q}}+{\frac {21}{\sqrt {q}}}-{\frac {18}{q^{3/2}}}+{\frac {12}{q^{5/2}}}-{\frac {8}{q^{7/2}}}+{\frac {3}{q^{9/2}}}-{\frac {1}{q^{11/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-1}+3az^{5}-4z^{5}a^{-1}+z^{5}a^{-3}-3a^{3}z^{3}+8az^{3}-8z^{3}a^{-1}+2z^{3}a^{-3}+a^{5}z-5a^{3}z+6az-6za^{-1}+2za^{-3}+a^{5}z^{-1}-a^{3}z^{-1}$ (db)
Kauffman polynomial	$-2a^{2}z^{10}-2z^{10}-4a^{3}z^{9}-12az^{9}-8z^{9}a^{-1}-3a^{4}z^{8}-6a^{2}z^{8}-14z^{8}a^{-2}-17z^{8}-a^{5}z^{7}+9a^{3}z^{7}+25az^{7}+z^{7}a^{-1}-14z^{7}a^{-3}+10a^{4}z^{6}+32a^{2}z^{6}+22z^{6}a^{-2}-9z^{6}a^{-4}+53z^{6}+4a^{5}z^{5}+a^{3}z^{5}-az^{5}+26z^{5}a^{-1}+20z^{5}a^{-3}-4z^{5}a^{-5}-10a^{4}z^{4}-32a^{2}z^{4}-10z^{4}a^{-2}+7z^{4}a^{-4}-z^{4}a^{-6}-40z^{4}-6a^{5}z^{3}-11a^{3}z^{3}-14az^{3}-21z^{3}a^{-1}-11z^{3}a^{-3}+z^{3}a^{-5}+2a^{4}z^{2}+9a^{2}z^{2}+2z^{2}a^{-2}-2z^{2}a^{-4}+11z^{2}+4a^{5}z+5a^{3}z+4az+5za^{-1}+2za^{-3}+a^{4}-a^{5}z^{-1}-a^{3}z^{-1}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 12                       1 -1 10                     3   3 8                   6 1   -5 6                 9 3     6 4               10 6       -4 2             11 9         2 0           11 11           0 -2         7 10             -3 -4       5 11               6 -6     3 7                 -4 -8   1 6                   5 -10   2                     -2 -12 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=0$ $i=2$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{3}$ $r=-3$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-2$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=-1$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=0$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{11}$ $r=1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=2$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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