L11n173

L11n172

L11n174

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n173 at Knotilus!

Link Presentations

[edit Notes on L11n173's Link Presentations]

Planar diagram presentation	X8192 X10,4,11,3 X12,7,13,8 X15,7,16,22 X14,6,15,5 X6,14,1,13 X21,17,22,16 X18,10,19,9 X20,11,21,12 X4,18,5,17 X2,19,3,20
Gauss code	{1, -11, 2, -10, 5, -6}, {3, -1, 8, -2, 9, -3, 6, -5, -4, 7, 10, -8, 11, -9, -7, 4}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

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

Integral Khovanov Homology (db, data source)

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

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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