L11a425

L11a424

L11a426

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a425 at Knotilus!

Link Presentations

[edit Notes on L11a425's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {2uv^{2}w^{2}-3uv^{2}w+uv^{2}-5uvw^{2}+7uvw-3uv+3uw^{2}-5uw+2u-2v^{2}w^{2}+5v^{2}w-3v^{2}+3vw^{2}-7vw+5v-w^{2}+3w-2}{{\sqrt {u}}vw}}$ (db)
Jones polynomial	$-q^{8}+4q^{7}-9q^{6}+14q^{5}-18q^{4}+21q^{3}-19q^{2}+17q-11+7q^{-1}-2q^{-2}+q^{-3}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-z^{4}a^{-6}-z^{2}a^{-6}-a^{-6}+z^{6}a^{-4}+2z^{4}a^{-4}+3z^{2}a^{-4}+2a^{-4}+z^{6}a^{-2}+z^{4}a^{-2}+a^{2}z^{2}+a^{2}z^{-2}+a^{-2}z^{-2}+2a^{2}+a^{-2}-2z^{4}-4z^{2}-2z^{-2}-4$ (db)
Kauffman polynomial	$z^{5}a^{-9}-z^{3}a^{-9}+4z^{6}a^{-8}-5z^{4}a^{-8}+z^{2}a^{-8}+8z^{7}a^{-7}-13z^{5}a^{-7}+7z^{3}a^{-7}-2za^{-7}+9z^{8}a^{-6}-14z^{6}a^{-6}+9z^{4}a^{-6}-5z^{2}a^{-6}+2a^{-6}+5z^{9}a^{-5}+4z^{7}a^{-5}-21z^{5}a^{-5}+17z^{3}a^{-5}-4za^{-5}+z^{10}a^{-4}+16z^{8}a^{-4}-39z^{6}a^{-4}+37z^{4}a^{-4}-18z^{2}a^{-4}+4a^{-4}+8z^{9}a^{-3}-7z^{7}a^{-3}-6z^{5}a^{-3}+5z^{3}a^{-3}+z^{10}a^{-2}+10z^{8}a^{-2}+a^{2}z^{6}-25z^{6}a^{-2}-4a^{2}z^{4}+21z^{4}a^{-2}+6a^{2}z^{2}-7z^{2}a^{-2}+a^{2}z^{-2}+a^{-2}z^{-2}-4a^{2}-2a^{-2}+3z^{9}a^{-1}+2az^{7}-z^{7}a^{-1}-4az^{5}-3z^{5}a^{-1}-4z^{3}a^{-1}+4az+6za^{-1}-2az^{-1}-2a^{-1}z^{-1}+3z^{8}-3z^{6}-6z^{4}+11z^{2}+2z^{-2}-7$ (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 6 7 χ 17                       1 -1 15                     3   3 13                   6 1   -5 11                 8 3     5 9               10 6       -4 7             11 8         3 5           9 11           2 3         8 10             -2 1       5 11               6 -1     2 6                 -4 -3   1 6                   5 -5   1                     -1 -7 1                       1

Integral Khovanov Homology (db, data source)

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

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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