L11a137

L11a136

L11a138

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a137 at Knotilus!

Link Presentations

[edit Notes on L11a137's Link Presentations]

Planar diagram presentation	X6172 X2,11,3,12 X14,3,15,4 X12,5,13,6 X22,13,5,14 X4,21,1,22 X20,16,21,15 X16,8,17,7 X18,10,19,9 X8,18,9,17 X10,20,11,19
Gauss code	{1, -2, 3, -6}, {4, -1, 8, -10, 9, -11, 2, -4, 5, -3, 7, -8, 10, -9, 11, -7, 6, -5}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {2t(1)t(2)^{5}-2t(2)^{5}-5t(1)t(2)^{4}+6t(2)^{4}+6t(1)t(2)^{3}-6t(2)^{3}-6t(1)t(2)^{2}+6t(2)^{2}+6t(1)t(2)-5t(2)-2t(1)+2}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-4q^{9/2}+{\frac {3}{q^{9/2}}}+8q^{7/2}-{\frac {6}{q^{7/2}}}-12q^{5/2}+{\frac {9}{q^{5/2}}}+16q^{3/2}-{\frac {15}{q^{3/2}}}+q^{11/2}-{\frac {1}{q^{11/2}}}-17{\sqrt {q}}+{\frac {16}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-az^{7}-z^{7}a^{-1}+a^{3}z^{5}-4az^{5}-3z^{5}a^{-1}+z^{5}a^{-3}+3a^{3}z^{3}-6az^{3}-z^{3}a^{-1}+2z^{3}a^{-3}+2a^{3}z-6az+2za^{-1}+2a^{3}z^{-1}-3az^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-2a^{2}z^{10}-2z^{10}-4a^{3}z^{9}-10az^{9}-6z^{9}a^{-1}-3a^{4}z^{8}-9z^{8}a^{-2}-6z^{8}-a^{5}z^{7}+15a^{3}z^{7}+33az^{7}+7z^{7}a^{-1}-10z^{7}a^{-3}+12a^{4}z^{6}+17a^{2}z^{6}+11z^{6}a^{-2}-8z^{6}a^{-4}+24z^{6}+4a^{5}z^{5}-18a^{3}z^{5}-39az^{5}-z^{5}a^{-1}+12z^{5}a^{-3}-4z^{5}a^{-5}-13a^{4}z^{4}-23a^{2}z^{4}+8z^{4}a^{-4}-z^{4}a^{-6}-19z^{4}-4a^{5}z^{3}+11a^{3}z^{3}+24az^{3}+4z^{3}a^{-1}-3z^{3}a^{-3}+2z^{3}a^{-5}+3a^{4}z^{2}+10a^{2}z^{2}-2z^{2}a^{-2}-2z^{2}a^{-4}+7z^{2}+a^{5}z-5a^{3}z-8az-2za^{-1}-3a^{2}-a^{-2}-3+2a^{3}z^{-1}+3az^{-1}+a^{-1}z^{-1}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 12                       1 -1 10                     3   3 8                   5 1   -4 6                 7 3     4 4               9 5       -4 2             8 7         1 0           9 10           1 -2         6 7             -1 -4       3 9               6 -6     3 6                 -3 -8   1 4                   3 -10   2                     -2 -12 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=0$ $i=2$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{3}$ $r=-3$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-2$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-1$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=0$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{8}$ $r=1$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=2$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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