# K11n42

## Contents

 (Knotscape image) See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n42's page at Knotilus! Visit K11n42's page at the original Knot Atlas! K11n42 is the mirror of the "Kinoshita-Terasaka" knot; it is a mutant of the (mirror of the) Conway knot K11n34. See also Heegaard Floer Knot Homology.

K11n42 is not k-colourable for any k. See The Determinant and the Signature.

 Knot K11n42. A graph, knot K11n42. A part of a knot and a part of a graph.

### Knot presentations

 Planar diagram presentation X4251 X8493 X12,5,13,6 X2837 X9,18,10,19 X11,21,12,20 X6,13,7,14 X15,10,16,11 X17,22,18,1 X19,15,20,14 X21,16,22,17 Gauss code 1, -4, 2, -1, 3, -7, 4, -2, -5, 8, -6, -3, 7, 10, -8, 11, -9, 5, -10, 6, -11, 9 Dowker-Thistlethwaite code 4 8 12 2 -18 -20 6 -10 -22 -14 -16

### Three dimensional invariants

 Symmetry type Chiral Unknotting number 1 3-genus 2 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n42/ThurstonBennequinNumber Hyperbolic Volume 11.2191 A-Polynomial See Data:K11n42/A-polynomial

### Four dimensional invariants

 Smooth 4 genus Missing Topological 4 genus Missing Concordance genus 0 Rasmussen s-Invariant 0

### Polynomial invariants

 Alexander polynomial 1 Conway polynomial 1 2nd Alexander ideal (db, data sources) {1} Determinant and Signature { 1, 0 } Jones polynomial −q4 + 2q3−2q2 + 2q + q−2−2q−3 + 2q−4−2q−5 + q−6 HOMFLY-PT polynomial (db, data sources) −a2z6 + z6 + a4z4−6a2z4−z4a−2 + 6z4 + 3a4z2−11a2z2−3z2a−2 + 11z2 + 2a4−6a2−2a−2 + 7 Kauffman polynomial (db, data sources) az9 + z9a−1 + a4z8 + 2a2z8 + 2z8a−2 + 3z8 + 2a5z7 + 2a3z7−5az7−4z7a−1 + z7a−3 + a6z6−4a4z6−14a2z6−11z6a−2−20z6−9a5z5−12a3z5−2z5a−1−5z5a−3−4a6z4 + 2a4z4 + 26a2z4 + 16z4a−2 + 36z4 + 9a5z3 + 16a3z3 + 12az3 + 11z3a−1 + 6z3a−3 + 3a6z2−2a4z2−20a2z2−9z2a−2−24z2−3a5z−7a3z−7az−5za−1−2za−3 + 2a4 + 6a2 + 2a−2 + 7 The A2 invariant q18 + q14−q12−q10−q8−2q6 + q4 + 3 + 2q−2 + q−4 + q−6−q−8−q−12 The G2 invariant Data:K11n42/QuantumInvariant/G2/1,0

### "Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {0_1, K11n34,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11n34,}

### Vassiliev invariants

 V2 and V3: (0, 2)
V2,1 through V6,9:
 V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 0 16 0 −16 0 0 $-\frac{128}{3}$ $-\frac{128}{3}$ −16 0 128 0 0 312 $\frac{32}{3}$ 176 8 8

V2,1 through V6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V2 and V3.

### 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 = 0 is the signature of K11n42. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
-6-5-4-3-2-1012345χ
9           1-1
7          1 1
5         11 0
3       121  0
1      211   2
-1     132    0
-3    221     1
-5   111      -1
-7  121       0
-9 11         0
-11 1          -1
-131           1
Integral Khovanov Homology
 $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ i = −3 i = −1 i = 1 r = −6 ${\mathbb Z}$ r = −5 ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ r = −4 ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ r = −3 ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ r = −2 ${\mathbb Z}$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ r = −1 ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ r = 0 ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ r = 1 ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ r = 2 ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ r = 3 ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ r = 4 ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ 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.

Read me first: Modifying Knot Pages.

See/edit the Hoste-Thistlethwaite Knot Page master template (intermediate).

See/edit the Hoste-Thistlethwaite_Splice_Base (expert).

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