L11a437

L11a436

L11a438

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a437 at Knotilus!

Link Presentations

[edit Notes on L11a437's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(v-1)(w-1)\left(3uv-2u+2v^{2}-3v\right)}{{\sqrt {u}}v^{3/2}{\sqrt {w}}}}$ (db)
Jones polynomial	$-q^{-10}+3q^{-9}-6q^{-8}+9q^{-7}-10q^{-6}+13q^{-5}-12q^{-4}+11q^{-3}-7q^{-2}+q+5q^{-1}-2$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-a^{10}+3a^{8}z^{2}+2a^{8}-2a^{6}z^{4}-2a^{6}z^{2}+a^{6}z^{-2}-2a^{4}z^{4}-2a^{4}z^{2}-2a^{4}z^{-2}-3a^{4}-a^{2}z^{4}+a^{2}z^{-2}+a^{2}+z^{2}+1$ (db)
Kauffman polynomial	$z^{7}a^{11}-4z^{5}a^{11}+5z^{3}a^{11}-2za^{11}+3z^{8}a^{10}-12z^{6}a^{10}+14z^{4}a^{10}-7z^{2}a^{10}+2a^{10}+3z^{9}a^{9}-8z^{7}a^{9}-2z^{5}a^{9}+11z^{3}a^{9}-4za^{9}+z^{10}a^{8}+5z^{8}a^{8}-30z^{6}a^{8}+38z^{4}a^{8}-18z^{2}a^{8}+4a^{8}+6z^{9}a^{7}-17z^{7}a^{7}+10z^{5}a^{7}-z^{3}a^{7}+z^{10}a^{6}+5z^{8}a^{6}-21z^{6}a^{6}+19z^{4}a^{6}-3z^{2}a^{6}+a^{6}z^{-2}-3a^{6}+3z^{9}a^{5}-5z^{7}a^{5}+6z^{5}a^{5}-10z^{3}a^{5}+8za^{5}-2a^{5}z^{-1}+3z^{8}a^{4}-9z^{4}a^{4}+13z^{2}a^{4}+2a^{4}z^{-2}-8a^{4}+3z^{7}a^{3}-5z^{3}a^{3}+6za^{3}-2a^{3}z^{-1}+3z^{6}a^{2}-3z^{4}a^{2}+3z^{2}a^{2}+a^{2}z^{-2}-3a^{2}+2z^{5}a-2z^{3}a+z^{4}-2z^{2}+1$ (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   \ -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 3                       1 1 1                     1   -1 -1                   4 1   3 -3                 5 3     -2 -5               6 2       4 -7             6 5         -1 -9           7 6           1 -11         5 8             3 -13       4 5               -1 -15     2 5                 3 -17   1 4                   -3 -19   2                     2 -21 1                       -1

Integral Khovanov Homology (db, data source)

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

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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