# K11a65

## Contents

 (Knotscape image) See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a65's page at Knotilus! Visit K11a65's page at the original Knot Atlas!

### Knot presentations

 Planar diagram presentation X4251 X8394 X16,5,17,6 X10,8,11,7 X2,9,3,10 X22,11,1,12 X20,13,21,14 X18,15,19,16 X6,17,7,18 X14,19,15,20 X12,21,13,22 Gauss code 1, -5, 2, -1, 3, -9, 4, -2, 5, -4, 6, -11, 7, -10, 8, -3, 9, -8, 10, -7, 11, -6 Dowker-Thistlethwaite code 4 8 16 10 2 22 20 18 6 14 12
A Braid Representative

### Three dimensional invariants

 Symmetry type Reversible Unknotting number 2 3-genus 2 Bridge index 2 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a65/ThurstonBennequinNumber Hyperbolic Volume 10.6865 A-Polynomial See Data:K11a65/A-polynomial

### Four dimensional invariants

 Smooth 4 genus Missing Topological 4 genus Missing Concordance genus 2 Rasmussen s-Invariant 2

### Polynomial invariants

 Alexander polynomial −3t2 + 15t−23 + 15t−1−3t−2 Conway polynomial −3z4 + 3z2 + 1 2nd Alexander ideal (db, data sources) {1} Determinant and Signature { 59, -2 } Jones polynomial q−2 + 4q−1−6q−2 + 8q−3−9q−4 + 9q−5−7q−6 + 6q−7−4q−8 + 2q−9−q−10 HOMFLY-PT polynomial (db, data sources) −a10 + 2z2a8 + a8−z4a6−z4a4 + z2a4 + a4−z4a2−z2a2−a2 + z2 + 1 Kauffman polynomial (db, data sources) z7a11−5z5a11 + 7z3a11−3za11 + 2z8a10−9z6a10 + 11z4a10−5z2a10 + a10 + 2z9a9−7z7a9 + 3z5a9 + 4z3a9−2za9 + z10a8−z8a8−7z6a8 + 9z4a8−3z2a8 + a8 + 4z9a7−15z7a7 + 17z5a7−8z3a7 + za7 + z10a6−z8a6−2z6a6 + z4a6 + z2a6 + 2z9a5−5z7a5 + 7z5a5−4z3a5 + 2z8a4−2z6a4 + 2z4a4−2z2a4 + a4 + 2z7a3−2z3a3 + za3 + 2z6a2−3z2a2 + a2 + 2z5a−3z3a + za + z4−2z2 + 1 The A2 invariant Data:K11a65/QuantumInvariant/A2/1,0 The G2 invariant Data:K11a65/QuantumInvariant/G2/1,0

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

Same Alexander/Conway Polynomial: {}

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

### Vassiliev invariants

 V2 and V3: (3, -9)
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 12 −72 72 366 58 −864 −2128 −320 −392 288 2592 4392 696 $\frac{127231}{10}$ $-\frac{10786}{15}$ $\frac{91862}{15}$ $\frac{1793}{6}$ $\frac{8351}{10}$

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 = -2 is the signature of K11a65. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
-9-8-7-6-5-4-3-2-1012χ
3           11
1          1 -1
-1         31 2
-3        42  -2
-5       42   2
-7      54    -1
-9     44     0
-11    35      2
-13   34       -1
-15  13        2
-17 13         -2
-19 1          1
-211           -1
Integral Khovanov Homology
 $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ i = −3 i = −1 r = −9 ${\mathbb Z}$ r = −8 ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ r = −7 ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ r = −6 ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ r = −5 ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ r = −4 ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ r = −3 ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ r = −2 ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ r = −1 ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ r = 0 ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ r = 1 ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ r = 2 ${\mathbb Z}_2$ ${\mathbb Z}$

### 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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