L11a475

L11a474

L11a476

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a475 at Knotilus!

Link Presentations

[edit Notes on L11a475's Link Presentations]

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

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {(t(2)-1)(t(3)-1)^{2}(t(3)t(1)-3t(1)-3t(3)+1)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{5}+4q^{4}-9q^{3}+15q^{2}-19q+21-20q^{-1}+18q^{-2}-11q^{-3}+7q^{-4}-2q^{-5}+q^{-6}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{6}z^{-2}+a^{6}-3a^{4}z^{2}-2a^{4}z^{-2}-z^{2}a^{-4}-4a^{4}+3a^{2}z^{4}+2z^{4}a^{-2}+4a^{2}z^{2}+a^{2}z^{-2}+z^{2}a^{-2}+3a^{2}-z^{6}-z^{4}-z^{2}$ (db)
Kauffman polynomial	$a^{2}z^{10}+z^{10}+3a^{3}z^{9}+8az^{9}+5z^{9}a^{-1}+3a^{4}z^{8}+11a^{2}z^{8}+9z^{8}a^{-2}+17z^{8}+2a^{5}z^{7}-3az^{7}+7z^{7}a^{-1}+8z^{7}a^{-3}+a^{6}z^{6}-3a^{4}z^{6}-27a^{2}z^{6}-11z^{6}a^{-2}+4z^{6}a^{-4}-38z^{6}-4a^{5}z^{5}-7a^{3}z^{5}-18az^{5}-28z^{5}a^{-1}-12z^{5}a^{-3}+z^{5}a^{-5}-4a^{6}z^{4}-5a^{4}z^{4}+22a^{2}z^{4}+3z^{4}a^{-2}-5z^{4}a^{-4}+31z^{4}+4a^{3}z^{3}+18az^{3}+22z^{3}a^{-1}+7z^{3}a^{-3}-z^{3}a^{-5}+6a^{6}z^{2}+9a^{4}z^{2}-7a^{2}z^{2}-2z^{2}a^{-2}+2z^{2}a^{-4}-14z^{2}+4a^{5}z+2a^{3}z-6az-6za^{-1}-2za^{-3}-4a^{6}-6a^{4}-a^{2}+a^{-2}+3-2a^{5}z^{-1}-2a^{3}z^{-1}+a^{6}z^{-2}+2a^{4}z^{-2}+a^{2}z^{-2}$ (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   \ -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 11                       1 -1 9                     3   3 7                   6 1   -5 5                 9 3     6 3               10 6       -4 1             11 9         2 -1           11 12           1 -3         7 9             -2 -5       4 11               7 -7     3 7                 -4 -9   1 6                   5 -11   1                     -1 -13 1                       1

Integral Khovanov Homology (db, data source)

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

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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