L7n1

 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L7n1 at Knotilus! L7n1 is ${\displaystyle 7_{7}^{2}}$ in the Rolfsen table of links.

 Planar diagram presentation X6172 X12,7,13,8 X4,13,1,14 X5,10,6,11 X3849 X9,14,10,5 X11,2,12,3 Gauss code {1, 7, -5, -3}, {-4, -1, 2, 5, -6, 4, -7, -2, 3, 6}

Polynomial invariants

 Multivariable Alexander Polynomial (in ${\displaystyle u}$, ${\displaystyle v}$, ${\displaystyle w}$, ...) ${\displaystyle {\frac {-uv^{3}-1}{{\sqrt {u}}v^{3/2}}}}$ (db) Jones polynomial ${\displaystyle -{\frac {1}{q^{5/2}}}-{\frac {1}{q^{9/2}}}-{\frac {1}{q^{13/2}}}+{\frac {1}{q^{15/2}}}}$ (db) Signature -5 (db) HOMFLY-PT polynomial ${\displaystyle -a^{9}z^{-1}+a^{7}z^{3}+4a^{7}z+3a^{7}z^{-1}-a^{5}z^{5}-5a^{5}z^{3}-6a^{5}z-2a^{5}z^{-1}}$ (db) Kauffman polynomial ${\displaystyle a^{10}+a^{9}z-a^{9}z^{-1}+a^{8}z^{4}-4a^{8}z^{2}+3a^{8}+a^{7}z^{5}-5a^{7}z^{3}+7a^{7}z-3a^{7}z^{-1}+a^{6}z^{4}-4a^{6}z^{2}+3a^{6}+a^{5}z^{5}-5a^{5}z^{3}+6a^{5}z-2a^{5}z^{-1}}$ (db)

Khovanov Homology

The coefficients of the monomials ${\displaystyle t^{r}q^{j}}$ are shown, along with their alternating sums ${\displaystyle \chi }$ (fixed ${\displaystyle j}$, alternation over ${\displaystyle r}$).
 \ r \ j \
-5-4-3-2-10χ
-4     11
-6     11
-8   1  1
-10 1    1
-12 21   1
-14      0
-161     -1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-6}$ ${\displaystyle i=-4}$ ${\displaystyle i=-2}$ ${\displaystyle r=-5}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-1}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }}$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory. See A Sample KnotTheory Session.

Read me first: Modifying Knot Pages

See/edit the Link Page master template (intermediate).