K11a329

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K11a328.gif

K11a328

K11a330.gif

K11a330

Contents

K11a329.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a329 at Knotilus!



Knot presentations

Planar diagram presentation X6271 X12,4,13,3 X18,6,19,5 X22,8,1,7 X14,10,15,9 X4,12,5,11 X20,14,21,13 X8,16,9,15 X2,18,3,17 X10,20,11,19 X16,22,17,21
Gauss code 1, -9, 2, -6, 3, -1, 4, -8, 5, -10, 6, -2, 7, -5, 8, -11, 9, -3, 10, -7, 11, -4
Dowker-Thistlethwaite code 6 12 18 22 14 4 20 8 2 10 16
A Braid Representative
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A Morse Link Presentation K11a329 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{2,3\}
3-genus 2
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a329/ThurstonBennequinNumber
Hyperbolic Volume 17.4305
A-Polynomial See Data:K11a329/A-polynomial

[edit Notes for K11a329's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a329's four dimensional invariants]

Polynomial invariants

Alexander polynomial 11 t^2-36 t+51-36 t^{-1} +11 t^{-2}
Conway polynomial 11 z^4+8 z^2+1
2nd Alexander ideal (db, data sources) \left\{2,t^2+t+1\right\}
Determinant and Signature { 145, 4 }
Jones polynomial -q^{13}+4 q^{12}-9 q^{11}+14 q^{10}-20 q^9+23 q^8-23 q^7+21 q^6-15 q^5+10 q^4-4 q^3+q^2
HOMFLY-PT polynomial (db, data sources) z^4 a^{-4} +4 z^4 a^{-6} +5 z^4 a^{-8} +z^4 a^{-10} +5 z^2 a^{-6} +8 z^2 a^{-8} -4 z^2 a^{-10} -z^2 a^{-12} + a^{-6} +4 a^{-8} -5 a^{-10} + a^{-12}
Kauffman polynomial (db, data sources) 3 z^{10} a^{-10} +3 z^{10} a^{-12} +10 z^9 a^{-9} +16 z^9 a^{-11} +6 z^9 a^{-13} +16 z^8 a^{-8} +17 z^8 a^{-10} +5 z^8 a^{-12} +4 z^8 a^{-14} +15 z^7 a^{-7} -5 z^7 a^{-9} -38 z^7 a^{-11} -17 z^7 a^{-13} +z^7 a^{-15} +10 z^6 a^{-6} -26 z^6 a^{-8} -59 z^6 a^{-10} -36 z^6 a^{-12} -13 z^6 a^{-14} +4 z^5 a^{-5} -19 z^5 a^{-7} -24 z^5 a^{-9} +13 z^5 a^{-11} +11 z^5 a^{-13} -3 z^5 a^{-15} +z^4 a^{-4} -9 z^4 a^{-6} +16 z^4 a^{-8} +49 z^4 a^{-10} +36 z^4 a^{-12} +13 z^4 a^{-14} +7 z^3 a^{-7} +21 z^3 a^{-9} +13 z^3 a^{-11} +2 z^3 a^{-13} +3 z^3 a^{-15} +5 z^2 a^{-6} -11 z^2 a^{-8} -20 z^2 a^{-10} -8 z^2 a^{-12} -4 z^2 a^{-14} -7 z a^{-9} -7 z a^{-11} -z a^{-13} -z a^{-15} - a^{-6} +4 a^{-8} +5 a^{-10} + a^{-12}
The A2 invariant Data:K11a329/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a329/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: (8, 21)
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
32 168 512 \frac{3472}{3} \frac{488}{3} 5376 8880 1504 1064 \frac{16384}{3} 14112 \frac{111104}{3} \frac{15616}{3} \frac{1049884}{15} \frac{46424}{15} \frac{1106776}{45} \frac{4916}{9} \frac{46204}{15}

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 t^rq^j are shown, along with their alternating sums \chi (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=4 is the signature of K11a329. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
01234567891011χ
27           1-1
25          3 3
23         61 -5
21        83  5
19       126   -6
17      118    3
15     1212     0
13    911      -2
11   612       6
9  49        -5
7  6         6
514          -3
31           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=3 i=5
r=0 {\mathbb Z} {\mathbb Z}
r=1 {\mathbb Z}^{4}
r=2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=3 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=4 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=5 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=6 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=7 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=8 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=9 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=10 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=11 {\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 Hoste-Thistlethwaite Knot Page master template (intermediate).

See/edit the Hoste-Thistlethwaite_Splice_Base (expert).

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K11a328.gif

K11a328

K11a330.gif

K11a330