# L11a312

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## Contents

 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a312's page at Knotilus. Visit L11a312's page at the original Knot Atlas.

### Link Presentations

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

### Polynomial invariants

 Multivariable Alexander Polynomial (in u, v, w, ...) $-\frac{(t(1)-1) (t(2)-1) \left(t(2)^2 t(1)^2-2 t(2) t(1)^2-2 t(2)^2 t(1)-2 t(1)-2 t(2)+1\right)}{t(1)^{3/2} t(2)^{3/2}}$ (db) Jones polynomial $-q^{5/2}+2 q^{3/2}-5 \sqrt{q}+\frac{7}{\sqrt{q}}-\frac{10}{q^{3/2}}+\frac{12}{q^{5/2}}-\frac{13}{q^{7/2}}+\frac{11}{q^{9/2}}-\frac{9}{q^{11/2}}+\frac{6}{q^{13/2}}-\frac{3}{q^{15/2}}+\frac{1}{q^{17/2}}$ (db) Signature -3 (db) HOMFLY-PT polynomial −z3a7−2za7 + 2z5a5 + 6z3a5 + 3za5−z7a3−4z5a3−5z3a3−3za3 + 2z5a + 7z3a + 5za + az−1−z3a−1−3za−1−a−1z−1 (db) Kauffman polynomial a10z4−a10z2 + 3a9z5−3a9z3 + 5a8z6−6a8z4 + 2a8z2 + 6a7z7−10a7z5 + 9a7z3−3a7z + 5a6z8−8a6z6 + 7a6z4−3a6z2 + 3a5z9−2a5z7−5a5z5 + 6a5z3−2a5z + a4z10 + 4a4z8−16a4z6 + 15a4z4−5a4z2 + 5a3z9−14a3z7 + 10a3z5−3a3z3 + a2z10 + a2z8−11a2z6 + 9a2z4 + 2az9−5az7 + z7a−1−3az5−5z5a−1 + 11az3 + 8z3a−1−6az + az−1−5za−1 + a−1z−1 + 2z8−8z6 + 8z4−z2−1 (db)

### Khovanov Homology

 The coefficients of the monomials trqj are shown, along with their alternating sums χ (fixed j, alternation over r). The squares with yellow highlighting are those on the "critical diagonals", where j−2r = s + 1 or j−2r = s−1, where s = -3 is the signature of L11a312. Nonzero entries off the critical diagonals (if any exist) are highlighted in red. Data:L11a312/KhovanovTable
Integral Khovanov Homology
 $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ i = −4 i = −2 r = −7 ${\mathbb Z}$ r = −6 ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ r = −5 ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ r = −4 ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ r = −3 ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ r = −2 ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ r = −1 ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ r = 0 ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{7}$ r = 1 ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ r = 2 ${\mathbb Z}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ r = 3 ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ r = 4 ${\mathbb Z}_2$ ${\mathbb Z}$

### 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).

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