L11n237

L11n236

L11n238

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n237 at Knotilus!

Link Presentations

[edit Notes on L11n237's Link Presentations]

Planar diagram presentation	X10,1,11,2 X3,14,4,15 X22,5,9,6 X6,9,7,10 X20,12,21,11 X18,14,19,13 X12,20,13,19 X7,16,8,17 X15,4,16,5 X17,8,18,1 X2,21,3,22
Gauss code	{1, -11, -2, 9, 3, -4, -8, 10}, {4, -1, 5, -7, 6, 2, -9, 8, -10, -6, 7, -5, 11, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {\left(t(2)t(1)^{2}-2t(1)^{2}-2t(2)+1\right)\left(t(1)t(2)^{2}+1\right)}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$-{\frac {1}{\sqrt {q}}}+{\frac {1}{q^{3/2}}}-{\frac {1}{q^{5/2}}}+{\frac {1}{q^{7/2}}}-{\frac {1}{q^{11/2}}}-{\frac {1}{q^{15/2}}}+{\frac {1}{q^{17/2}}}-{\frac {1}{q^{19/2}}}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$2a^{9}z+a^{9}z^{-1}-a^{7}z^{5}-6a^{7}z^{3}-7a^{7}z-a^{7}z^{-1}+a^{5}z^{7}+6a^{5}z^{5}+10a^{5}z^{3}+6a^{5}z-a^{3}z^{5}-5a^{3}z^{3}-5a^{3}z$ (db)
Kauffman polynomial	$-z^{5}a^{11}+4z^{3}a^{11}-2za^{11}-z^{6}a^{10}+4z^{4}a^{10}-2z^{2}a^{10}-z^{7}a^{9}+6z^{5}a^{9}-10z^{3}a^{9}+7za^{9}-a^{9}z^{-1}-z^{6}a^{8}+6z^{4}a^{8}-6z^{2}a^{8}+a^{8}-2z^{7}a^{7}+13z^{5}a^{7}-21z^{3}a^{7}+9za^{7}-a^{7}z^{-1}-z^{8}a^{6}+6z^{6}a^{6}-8z^{4}a^{6}+2z^{2}a^{6}-2z^{7}a^{5}+12z^{5}a^{5}-17z^{3}a^{5}+5za^{5}-z^{8}a^{4}+6z^{6}a^{4}-10z^{4}a^{4}+6z^{2}a^{4}-z^{7}a^{3}+6z^{5}a^{3}-10z^{3}a^{3}+5za^{3}$ (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 χ 0                     1 1 -2                       0 -4               1 2 1   0 -6               1 1     0 -8           2 2 1       -1 -10         1 1 1         1 -12       1 3 1           1 -14     1 1 1             1 -16     1 1               0 -18 1 1                   0 -20 1                     1

Integral Khovanov Homology (db, data source)

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

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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