L11a537

L11a536

L11a538

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a537 at Knotilus!

Link Presentations

[edit Notes on L11a537's Link Presentations]

Planar diagram presentation	X6172 X2536 X18,12,19,11 X10,3,11,4 X4,9,1,10 X8,18,5,17 X16,8,17,7 X14,16,9,15 X22,20,15,19 X20,13,21,14 X12,21,13,22
Gauss code	{1, -2, 4, -5}, {2, -1, 7, -6}, {5, -4, 3, -11, 10, -8}, {8, -7, 6, -3, 9, -10, 11, -9}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {-t(1)t(4)^{2}t(3)^{2}+t(1)t(2)t(4)^{2}t(3)^{2}-2t(2)t(4)^{2}t(3)^{2}+t(4)^{2}t(3)^{2}-t(1)t(3)^{2}-t(2)t(3)^{2}+2t(1)t(4)t(3)^{2}-t(1)t(2)t(4)t(3)^{2}+3t(2)t(4)t(3)^{2}-2t(4)t(3)^{2}+t(3)^{2}+2t(1)t(4)^{2}t(3)-2t(1)t(2)t(4)^{2}t(3)+3t(2)t(4)^{2}t(3)-t(4)^{2}t(3)+3t(1)t(3)-t(1)t(2)t(3)+2t(2)t(3)-4t(1)t(4)t(3)+2t(1)t(2)t(4)t(3)-4t(2)t(4)t(3)+2t(4)t(3)-2t(3)-t(1)t(4)^{2}+t(1)t(2)t(4)^{2}-t(2)t(4)^{2}-2t(1)+t(1)t(2)-t(2)+3t(1)t(4)-2t(1)t(2)t(4)+2t(2)t(4)-t(4)+1}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)t(4)}}$ (db)
Jones polynomial	$q^{11/2}-4q^{9/2}+9q^{7/2}-14q^{5/2}+17q^{3/2}-21{\sqrt {q}}+{\frac {17}{\sqrt {q}}}-{\frac {17}{q^{3/2}}}+{\frac {10}{q^{5/2}}}-{\frac {7}{q^{7/2}}}+{\frac {2}{q^{9/2}}}-{\frac {1}{q^{11/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-1}+3az^{5}-4z^{5}a^{-1}+z^{5}a^{-3}-3a^{3}z^{3}+10az^{3}-8z^{3}a^{-1}+2z^{3}a^{-3}+a^{5}z-8a^{3}z+14az-9za^{-1}+2za^{-3}+2a^{5}z^{-1}-8a^{3}z^{-1}+11az^{-1}-6a^{-1}z^{-1}+a^{-3}z^{-1}+a^{5}z^{-3}-3a^{3}z^{-3}+3az^{-3}-a^{-1}z^{-3}$ (db)
Kauffman polynomial	$z^{4}a^{-6}+a^{5}z^{7}-5a^{5}z^{5}+4z^{5}a^{-5}+10a^{5}z^{3}-z^{3}a^{-5}-a^{5}z^{-3}-10a^{5}z+5a^{5}z^{-1}+2a^{4}z^{8}-6a^{4}z^{6}+9z^{6}a^{-4}+3a^{4}z^{4}-8z^{4}a^{-4}+7a^{4}z^{2}+3z^{2}a^{-4}+3a^{4}z^{-2}-9a^{4}-a^{-4}+2a^{3}z^{9}+a^{3}z^{7}+13z^{7}a^{-3}-24a^{3}z^{5}-19z^{5}a^{-3}+45a^{3}z^{3}+13z^{3}a^{-3}-3a^{3}z^{-3}-35a^{3}z-6za^{-3}+14a^{3}z^{-1}+2a^{-3}z^{-1}+a^{2}z^{10}+7a^{2}z^{8}+11z^{8}a^{-2}-23a^{2}z^{6}-10z^{6}a^{-2}+6a^{2}z^{4}-10z^{4}a^{-2}+24a^{2}z^{2}+14z^{2}a^{-2}+6a^{2}z^{-2}-21a^{2}-6a^{-2}+7az^{9}+5z^{9}a^{-1}-az^{7}+12z^{7}a^{-1}-48az^{5}-52z^{5}a^{-1}+74az^{3}+53z^{3}a^{-1}-3az^{-3}-a^{-1}z^{-3}-49az-30za^{-1}+18az^{-1}+11a^{-1}z^{-1}+z^{10}+16z^{8}-36z^{6}+2z^{4}+28z^{2}+3z^{-2}-18$ (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                 8 3     5 4               10 7       -3 2             11 7         4 0           9 13           4 -2         8 8             0 -4       5 12               7 -6     2 5                 -3 -8   1 6                   5 -10   1                     -1 -12 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=0$ $i=2$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{2}$ $r=-3$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-2$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{8}$ $r=-1$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=0$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{11}$ $r=1$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=2$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{8}$ $r=3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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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