L11a360

L11a359

L11a361

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a360 at Knotilus!

Link Presentations

[edit Notes on L11a360's Link Presentations]

Planar diagram presentation	X12,1,13,2 X14,4,15,3 X22,14,11,13 X16,6,17,5 X18,8,19,7 X20,10,21,9 X2,11,3,12 X4,16,5,15 X6,18,7,17 X8,20,9,19 X10,22,1,21
Gauss code	{1, -7, 2, -8, 4, -9, 5, -10, 6, -11}, {7, -1, 3, -2, 8, -4, 9, -5, 10, -6, 11, -3}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {u^{4}v^{4}-u^{4}v^{3}-u^{3}v^{4}+u^{3}v^{3}-u^{3}v^{2}-u^{2}v^{3}+u^{2}v^{2}-u^{2}v-uv^{2}+uv-u-v+1}{u^{2}v^{2}}}$ (db)
Jones polynomial	$2q^{9/2}-2q^{7/2}+q^{5/2}-q^{3/2}+q^{25/2}-2q^{23/2}+2q^{21/2}-3q^{19/2}+3q^{17/2}-3q^{15/2}+3q^{13/2}-3q^{11/2}$ (db)
Signature	7 (db)
HOMFLY-PT polynomial	$-z^{9}a^{-7}+z^{7}a^{-5}-8z^{7}a^{-7}+z^{7}a^{-9}+7z^{5}a^{-5}-22z^{5}a^{-7}+6z^{5}a^{-9}+15z^{3}a^{-5}-25z^{3}a^{-7}+10z^{3}a^{-9}+10za^{-5}-11za^{-7}+4za^{-9}+a^{-5}z^{-1}-a^{-7}z^{-1}$ (db)
Kauffman polynomial	$-z^{10}a^{-6}-z^{10}a^{-8}-z^{9}a^{-5}-3z^{9}a^{-7}-2z^{9}a^{-9}+7z^{8}a^{-6}+5z^{8}a^{-8}-2z^{8}a^{-10}+8z^{7}a^{-5}+21z^{7}a^{-7}+11z^{7}a^{-9}-2z^{7}a^{-11}-15z^{6}a^{-6}-5z^{6}a^{-8}+8z^{6}a^{-10}-2z^{6}a^{-12}-22z^{5}a^{-5}-48z^{5}a^{-7}-18z^{5}a^{-9}+6z^{5}a^{-11}-2z^{5}a^{-13}+9z^{4}a^{-6}-3z^{4}a^{-8}-6z^{4}a^{-10}+4z^{4}a^{-12}-2z^{4}a^{-14}+25z^{3}a^{-5}+42z^{3}a^{-7}+11z^{3}a^{-9}-2z^{3}a^{-11}+2z^{3}a^{-13}-2z^{3}a^{-15}+2z^{2}a^{-6}+3z^{2}a^{-8}-z^{2}a^{-10}+z^{2}a^{-14}-z^{2}a^{-16}-11za^{-5}-13za^{-7}-3za^{-9}-za^{-11}+za^{-13}+za^{-15}-a^{-6}+a^{-5}z^{-1}+a^{-7}z^{-1}$ (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   \ -2 -1 0 1 2 3 4 5 6 7 8 9 χ 26                       1 -1 24                     1   1 22                   1 1   0 20                 2 1     1 18               2 2       0 16             1 1         0 14           2 2           0 12         1 1             0 10       1 2               1 8     1 1                 0 6   1 2                   1 4                         0 2 1                       1

Integral Khovanov Homology (db, data source)

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

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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