L7n2

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 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L7n2 at Knotilus! L7n2 is $7^2_8$ in the Rolfsen table of links.

Link Presentations

 Planar diagram presentation X6172 X12,7,13,8 X13,1,14,4 X5,10,6,11 X3849 X9,14,10,5 X2,12,3,11 Gauss code {1, -7, -5, 3}, {-4, -1, 2, 5, -6, 4, 7, -2, -3, 6}

Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(u-1) (v-1)}{\sqrt{u} \sqrt{v}}$ (db) Jones polynomial $-\frac{2}{\sqrt{q}}+\frac{1}{q^{3/2}}-\frac{2}{q^{5/2}}+\frac{1}{q^{7/2}}-\frac{1}{q^{9/2}}+\frac{1}{q^{11/2}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $a^5 (-z)-a^5 z^{-1} +a^3 z^3+3 a^3 z+3 a^3 z^{-1} -2 a z-2 a z^{-1}$ (db) Kauffman polynomial $a^6 z^4-3 a^6 z^2+a^6+a^5 z^5-3 a^5 z^3+2 a^5 z-a^5 z^{-1} +2 a^4 z^4-5 a^4 z^2+3 a^4+a^3 z^5-3 a^3 z^3+5 a^3 z-3 a^3 z^{-1} +a^2 z^4-2 a^2 z^2+3 a^2+3 a z-2 a z^{-1}$ (db)

Khovanov Homology

The coefficients of the monomials $t^rq^j$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$).
 \ r \ j \
-5-4-3-2-10χ
0     22
-2    121
-4   1  1
-6   1  1
-8 11   0
-10      0
-121     -1
Integral Khovanov Homology
 $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}^{2}$ ${\mathbb Z}^{2}$

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).

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