L11a481

L11a480

L11a482

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a481 at Knotilus!

Link Presentations

[edit Notes on L11a481's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {(w-1)\left(2uvw^{2}-4uvw+2uv-uw^{2}+3uw-4u-4vw^{2}+3vw-v+2w^{2}-4w+2\right)}{{\sqrt {u}}{\sqrt {v}}w^{3/2}}}$ (db)
Jones polynomial	$-q^{8}+5q^{7}-10q^{6}+15q^{5}-19q^{4}+21q^{3}-20q^{2}+17q-10+7q^{-1}-2q^{-2}+q^{-3}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{6}a^{-2}+z^{6}a^{-4}+z^{4}a^{-2}+z^{4}a^{-4}-z^{4}a^{-6}-2z^{4}+a^{2}z^{2}-z^{2}a^{-4}-4z^{2}+2a^{2}+a^{-2}-a^{-4}+a^{-6}-3+a^{2}z^{-2}+a^{-2}z^{-2}-2z^{-2}$ (db)
Kauffman polynomial	$z^{5}a^{-9}+5z^{6}a^{-8}-5z^{4}a^{-8}+10z^{7}a^{-7}-15z^{5}a^{-7}+4z^{3}a^{-7}+10z^{8}a^{-6}-11z^{6}a^{-6}-4z^{4}a^{-6}+6z^{2}a^{-6}-2a^{-6}+5z^{9}a^{-5}+8z^{7}a^{-5}-28z^{5}a^{-5}+18z^{3}a^{-5}-4za^{-5}+z^{10}a^{-4}+15z^{8}a^{-4}-24z^{6}a^{-4}-z^{4}a^{-4}+12z^{2}a^{-4}-2a^{-4}+7z^{9}a^{-3}+2z^{7}a^{-3}-30z^{5}a^{-3}+34z^{3}a^{-3}-12za^{-3}+z^{10}a^{-2}+8z^{8}a^{-2}+a^{2}z^{6}-13z^{6}a^{-2}-4a^{2}z^{4}+6a^{2}z^{2}+6z^{2}a^{-2}+a^{2}z^{-2}+a^{-2}z^{-2}-4a^{2}-2a^{-2}+2z^{9}a^{-1}+2az^{7}+6z^{7}a^{-1}-4az^{5}-22z^{5}a^{-1}+20z^{3}a^{-1}+4az-4za^{-1}-2az^{-1}-2a^{-1}z^{-1}+3z^{8}-4z^{6}-2z^{4}+6z^{2}+2z^{-2}-5$ (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                     4   4 13                   6 1   -5 11                 9 4     5 9               11 7       -4 7             10 8         2 5           10 11           1 3         7 10             -3 1       5 12               7 -1     2 5                 -3 -3     5                   5 -5 1 2                     -1 -7 1                       1

Integral Khovanov Homology (db, data source)

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

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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